Muhammad Usman

Usman Zabit
email: usman.zabit@seecs.nust.edu.pk

Olivier D Bernal

Gulistan Raja

Blind identification of occurrence of multi-modality in laser-feedback-based self-mixing sensor

Self-mixing interferometry (SMI) is an attractive sensing scheme which typically relies on mono-modal operation of employed laser diode. However, change in laser modality can occur due to change in operating conditions. So, detection of occurrence of multi-modality in SMI signals is necessary to avoid erroneous metric measurements. Typically, processing of multi-modal SMI signals is a difficult task due to the diverse and complex nature of such signals. However, the proposed techniques can significantly ease this task by identifying the modal state of SMI signals with 100% success rate, so that interferometric fringes can be correctly interpreted for metric sensing applications.

doi: 10.3788/COL202018.011201 SELF-MIXING (SM) or optical feedback (OF) interferometry [START_REF] Taimre | Laser feedback interferometry: A tutorial on the self-mixing effect for coherent sensing[END_REF][START_REF] Donati | Developing self-mixing interferometry for instrumentation and measurements[END_REF] is actively researched for vibration, angle [START_REF] Sun | Simultaneous measurement of vibration amplitude and rotation angle based on a single-channel laser selfmixing interferometer[END_REF] , frequency [START_REF] Wang | Resonant characteristics of microcantilever by using self-mixing interferometer based on phase reconstruction method[END_REF] , size [START_REF] Wang | Size measurement of nano-particles using self-mixing effect[END_REF] , range-finding [START_REF] Wang | Optimized design of laser range finding system using the self-mixing effect in a single-mode VCSEL[END_REF] , topographical [START_REF] Sun | Measurement of microsurface topography using a self-mixing optical configuration[END_REF] , and seismic applications [START_REF] Wu | Experimental study on acoustic-to-seismic landmine detection based on laser self-mixing interferometer[END_REF] due to the simple, low-cost, and miniaturized nature of SM sensors. In order to design low-cost SM sensors, usually commercial off the shelf (COTS) laser diodes (LD) are preferred. However, due to OF inside the active laser cavity, such low-cost monomodal LDs are prone to mode switching (as a function of operating conditions [START_REF] Ruiz-Llata | Self-mixing technique for vibration measurements in a laser diode with multiple modes created by optical feedback[END_REF][START_REF] Tucker | Effect of multiple transverse modes in self-mixing sensors based on vertical-cavity surface-emitting lasers[END_REF][START_REF] Zabit | Automatic Detection of Multi-modality in Self-Mixing Interferometer[END_REF] ) resulting in multi-modal SM signals in which more than one laser mode undergoes SM. As a consequence, each interferometric fringe can no more be assumed to correspond to a remote displacement of λ/2 (where λ is the wavelength of LD), because in case of bimodal or tri-modal SM, an individual SM fringe does not correspond to a displacement of λ/2 anymore (see Fig. 1) [START_REF] Ruiz-Llata | Self-mixing technique for vibration measurements in a laser diode with multiple modes created by optical feedback[END_REF] . Multi-modal SM has been used to potentially increase measurement resolution [START_REF] Tucker | Effect of multiple transverse modes in self-mixing sensors based on vertical-cavity surface-emitting lasers[END_REF] as well as to measure free spectral range of the laser [START_REF] Kliese | Solving selfmixing equations for arbitrary feedback levels: a concise algorithm[END_REF] . However, any unidentified switching of a mono-modal laser sensor to multi-modal operation can cause any unidentified severe measurement error due to incorrect fringe interpretation.

The objective of this Letter is to robustly identify the occurrence of multi-modal SM signals so that an alert can be raised to appropriately interpret SM fringe-count and/or SM operating conditions can be changed (e.g. by changing LD current [START_REF] Pham | Selfmixing sensing under strong feedback using multimode semiconductor lasers[END_REF] or amount of OF [START_REF] Bernal | Robust method of stabilization of optical feedback regime by using adaptive optics for a self-mixing micro-interferometer laser displacement sensor[END_REF] ) to revert back to monomodal SM operation [START_REF] Kliese | Solving selfmixing equations for arbitrary feedback levels: a concise algorithm[END_REF][START_REF] Bernal | Classification of laser self-mixing interferometric signal under moderate feedback[END_REF] for which algorithms exist yielding high accuracy measurements [START_REF] Gao | Improved method for estimation of multiple parameters in selfmixing interferometry[END_REF][START_REF] Ehtesham | Analysis and Implementation of a Direct Phase Unwrapping Method for Displacement Measurement using Self-Mixing Interferometry[END_REF][START_REF] Fan | Improving the measurement performance for a self-mixing interferometry-based displacement sensing system[END_REF][START_REF] Khan | Adaptive cancellation of parasitic vibrations affecting a self-mixing interferometric laser sensor[END_REF][START_REF] Huang | Piece-wise transition detection algorithm for a self-mixing displacement sensor[END_REF] .

SM based multi-modality is reported to occur due to variation in parameters such as LD-to-target distance [START_REF] Lv | Effect of external cavity length on self-mixing signals in a multilongitudinal-mode Fabry Perot laser diode[END_REF][START_REF] Zhao | Measurement of the Free Spectral Range of the laser cavity based on multilongitudinal mode Laser Self-mixing Vibrator[END_REF] , temperature [START_REF] Ruiz-Llata | Self-mixing technique for vibration measurements in a laser diode with multiple modes created by optical feedback[END_REF] , or LD current [START_REF] Tucker | Effect of multiple transverse modes in self-mixing sensors based on vertical-cavity surface-emitting lasers[END_REF][START_REF] Pham | Selfmixing sensing under strong feedback using multimode semiconductor lasers[END_REF] . Measurement of laser emission spectra confirmed existence of multiple laser modes undergoing SM signal [START_REF] Ruiz-Llata | Self-mixing technique for vibration measurements in a laser diode with multiple modes created by optical feedback[END_REF][START_REF] Pham | Selfmixing sensing under strong feedback using multimode semiconductor lasers[END_REF][START_REF] Keeley | Measurement of the emission spectrum of a semiconductor laser using laser-feedback interferometry[END_REF] for different laser sources such as Fabry-Perot LD [START_REF] Lv | Effect of external cavity length on self-mixing signals in a multilongitudinal-mode Fabry Perot laser diode[END_REF] , quantum cascade laser [START_REF] Keeley | Measurement of the emission spectrum of a semiconductor laser using laser-feedback interferometry[END_REF] , and VCSEL [START_REF] Tucker | Effect of multiple transverse modes in self-mixing sensors based on vertical-cavity surface-emitting lasers[END_REF] etc.

Recently, a method based on an artificial neural network was proposed to classify mono-, and multi-modal SM signals with success rate of 98.75% [START_REF] Zabit | Automatic Detection of Multi-Modality in Self-Mixing Interferometer[END_REF] . However, this neural network based approach requires hand-crafted feature engineering. Pertinent features (based on temporal and spatial characteristics of SM fringes) are extracted only after performing correct SM fringe detection, a task which is difficult to achieve for noisy, experimental SM signals even when only one mode undergoes SMI, as attested by the use of advanced detection methods based on Hilbert transform [START_REF] Arriaga | Real-time algorithm for versatile displacement sensors based on self-mixing interferometry[END_REF] , customized wavelet transform [START_REF] Bernal | Robust detection of non-regular interferometric fringes from a self-mixing displacement sensor using Bi-Wavelet transform[END_REF] , double-derivative [START_REF] Magnani | Self-mixing vibrometer with real-time digital signal elaboration[END_REF] and signal envelope tracking [START_REF] Siddiqui | All analog processing of speckle affected self-mixing interferometric signals[END_REF] etc. However, in this Letter, multi-modality of the SM signal is identified without using robust fringe detection by evaluating four different SM signal statistics under different noise, optical feedback strength, amplitude of target vibration, and laser modality conditions. Use of majority vote among the four techniques has provided 100% identification success rate.

Various mono-and multi-modal SM signals were acquired by using two different LDs, L637P5 by Oclaro ® and HL6501MG by Hitachi ® , one at a time. A polished metallic ring (mounted on a mechanical shaker, SF-9324 by PASCO ® ) was used as the remote vibrating target.

L637P5 LD has operating wavelength λ0 of 637 nm and threshold current Ith of 20 mA, emitting 5 mW optical power. HL6501MG LD has λ0 of 650 nm and Ith of 45 mA, providing 35 mW optical power. Each LD has a built-in photodiode through which SM signals were obtained. Different mono-and multi-modal SM signals were acquired under varying optical feedback and LD operating current (Ioper) conditions. Multi-modal SM signals were observed to occur when both the optical feedback coupling (by using the focusing lens) and Ioper/Ith (by using higher Ioper) well exceeded unity. Each of the proposed four different techniques for identification of SM multi-modality is detailed below.

Variance based technique (VBT) is based on the parameter varp-diff, which is a measure of peak to peak dynamic variation of an SM signal. Clearly, varp-diff should be generally greater for a multi-modal SM signal due to composition of different modes producing dissimilar multimodal fringes, as opposed to a mono-modal signal in which similar fringes occur (see Fig. 1). Consequently, larger variation in amplitude occurs in multi-modal signals as compared to mono-modal signals.

However, to perform VBT on normalized SM signal, two main phases are required: (1) customized local maxima detection and (2) estimation and analysis of varp-diff.

Customized local maxima detection is done by the following steps, which are also presented in Fig. 2.

1) First, inter-maxima separation (SMsep) is computed by using auto-correlation of SM signal. SMsep is indicative of the distance in between two consecutive maxima.

2) Then, the mean value of input SM signal, denoted by 𝑆𝑀 ̅̅̅̅ , is computed. VBT second phase steps (see Fig. 3) are detailed below:

1) Differentiation of amplitude values of detected maxima (mxdiff) is taken to determine peak to peak dynamic variations (varp-diff ).

2) varp-diff is determined by taking variance of mxdiff values.

3) A threshold value (thvr) of varp-diff is employed and is compared with varp-diff value of under-process SM signal to determine the modality of input signal. If thvr < varp-diff, then input SM signal is considered a multimodal signal else it is considered a mono-modal signal. Note that this threshold (as well as subsequent thresholds in other techniques) is set in the light of various simulation results obtained under varying optical feedback coupling, amplitude of target vibration, and noise conditions as detailed ahead. http://dx.doi.org/10.3788/COL202018.011201

𝑆𝑀 𝑠𝑘𝑤 = ∑ (𝑆𝑀 𝑖 -𝑆𝑀 ̅̅̅̅̅ ) 3 )/𝑁 𝑁 𝑖=1 𝑆 3 (2)
Thus, SMskw is determined, and is compared with the corresponding threshold value of skewness parameter (thskw) to ascertain the modality of input SM signal, Steps of this technique are also shown in Fig ( 4).

Skewness-kurtosis based technique (SKBT) is based on

the ratio (SMskur = SMkur/SMskw) of above-mentioned SM signal parameters. As both SMkur and SMskw detailed above are good indicators of multi-modality, so their ratio (SMskur) is also investigated for identifying multi-modality. (Note that to avoid division by values of SMskw approaching 0, all values of SMskw < 0.02 were set to 0.02 to plot SMskur in Fig. 5(d).) Absolute value of SMskur is compared with employed threshold value (thskur). If SMskur < thskur then under-process SM signal is considered multi-modal else it is considered mono-modal. Steps of SKBT are also shown in Fig ( 4).

Let us now discuss how the various threshold values, used in each of the four presented techniques, were set by performing simulations for a representative sample of SM signals, by using SM model [START_REF] Kliese | Solving selfmixing equations for arbitrary feedback levels: a concise algorithm[END_REF] under different optical feedback coupling (such as frequently encountered weak-, and moderate-optical feedback regime [START_REF] Taimre | Laser feedback interferometry: A tutorial on the self-mixing effect for coherent sensing[END_REF][START_REF] Donati | Developing self-mixing interferometry for instrumentation and measurements[END_REF] ), amplitude of target vibration in terms of λ0, and additive noise Furthermore, to ascertain impact of additive noise on chosen parameters, simulations for weak feedback regime (C = 0.1) and moderate feedback regime (C = 4) are also performed (see Table 1 andTable 2 respectively It can be observed from Table 1 and Fig. 6 that value of parameters such as varp-diff is decreasing significantly as SNR improves. Higher SNR values result in fewer local maxima generated by noise (and thus not genuine fringes) to be wrongly considered as fringe. Therefore, the calculation of varp-diff will not take them into account and hence varp-diff value will decrease.

Conduct of these simulations under different levels of noise, amplitude of target vibration, and OF coupling provides information about the expected range and worstcase value of proposed parameters, resulting in extraction of different threshold values (see Table 3). In order to determine the performance of proposed techniques, experimental dataset was tested to identify the modality of these SM signals by using threshold values of An analysis of misidentified signals led to the observation that misidentification by the proposed techniques occurred for different SM signals. So, majority voting (MV) based on results of the four techniques was undertaken (for each tested signal) resulting in 100% success rate. If lower number of parameters is used for the sake of reducing the complexity of the blind identification then 𝑅 𝑠 = 95% if VBT is not used while 𝑅 𝑠 = 92% if only VBT and SBT are used, inclusive of MV in both the cases. Some correctly identified mono-and multi-modal experimental SM signals are graphically presented in Fig. 7, and Fig. 8, respectively.

To conclude, an OF based LD can provide multi-modal SM signal in place of usually encountered mono-modal SM signal because of mode-hopping caused by change in operating conditions such as LD-to-target distance. This can cause misinterpretation of SM fringe-count, resulting in drastic increase in metric measurement error. To avoid this error, a continuous monitoring of SM signal is necessary, so that, as SM signal becomes multi-modal, it could be detected immediately and possibly reverted back to mono-modal behavior (e.g. by changing LD current or OF strength). In this Letter, different techniques, based on SM signal statistics, are evaluated for future continuous monitoring of emission modality of low-cost LD based SM sensor. These proposed techniques have been successfully tested on experimentally acquired mono-, and multi-modal SM signals with success rate of 85% (VBT), 75% (KBT), 91% (SBT), and 93% (SKBT). Importantly, use of majority voting among the four proposed techniques has provided 100% success rate of SM modality identification.

Fig. 1 (

 1 a-b) presents two multi-modal SM signals based on HL6501MG LD with Ioper/Ith ratio of 78mA/45mA=1.73, and 82mA/45mA=1.82 respectively under high optical feedback coupling. However, as optical feedback coupling was reduced (by de-focusing the lens) then mono-modal signal occurred even when Ioper/Ith was 1.73 (see Fig. 1 (c)). A dataset of 60 SM signals (30 monoand 30 multi-modal SM signals) is used to verify the performance of the proposed techniques using SM signal statistical parameters.

3 )

 3 Input SM signal (having N number of samples) is divided into 'n' intervals by using n = round (N / SMsep). 4) Then, local maxima of each interval is determined. 5) Valleys (SM signal portions with lower amplitude around local maxima) on left (lv) and right (rv) is determined, for each local maximum of every interval. 6) Valley-less maxima are discarded and maxima with both valleys are retained. 7) Finally, amplitude values of maxima (having both valleys) are compared with 𝑆𝑀 ̅̅̅̅ , and maxima with greater amplitude values are retained and considered as genuine maxima, while those with lower amplitude values are removed.

Fig. 2 .

 2 Fig.2.Flowchart of customized maxima detection technique

Fig. 1 .

 1 Fig. 1. Experimental multi-(a-b) and mono-modal (c) SM signals acquired under different optical feedback coupling and operating current conditions based on HL6501MG LD (Ith=45mA) with Ioper of (a-c) 78 mA, and (b) 82 mA.

Fig. 3 .

 3 Fig. 3. Flowchart of variance based technique (VBT).

(

  resulting in different signal to noise ratio (SNR) of SM signals) conditions. Evolution of different parameters with respect to C, and amplitude of target vibration in the absence of noise for mono-model operation can be observed from Fig. 5. It can be observed from Fig. 5(a) that varp-diff is always lower than 0.017 for mono-modal noiseless SM signals. In Fig. 5(b) SMkur increases with C which is expected since the more C increases the more asymmetric the SM fringes become. Regarding SMskw (see Fig. 5(c)), for low C values (close to one), SMskw is close to zero as positive and negative fringes are similar. Then, as 𝐶 increases, |𝑆𝑀 𝑠𝑘𝑤 | value tends to increase due to the increasing asymmetry between the positive and negative fringes of SM signal.

  ). Two weak-and moderate-feedback regime SM signals under different noise conditions (SNR = 10 dB and SNR = 40 dB) are graphically shown in Fig. 6 as well. Value of C = 4 is specifically chosen to perform noise analysis as it generally corresponds to the worst-case statistical parameter values.

Fig. 4 .

 4 Fig. 4. Steps of Kurtosis based technique (KBT), skewness based technique (SBT), and skewness-kurtosis based technique (SKBT).

Fig. 5 .

 5 Fig. 5. Evolution of parameters with respect to C and target vibration amplitude for noiseless mono-modal signal (a) varp-diff (b) SMkur, (c) SMskw, and (d) |SMskur|

Fig. 7 .

 7 Fig. 7. (a-b) Sample correctly identified experimental mono-modal SM signals.

Fig. 8 .

 8 Fig. 8. Sample correctly identified experimental (a) bimodal-, and (b) tri-modal-SM signals.

  , where Nmon and Nmul indicates number of tested mono-, and multi-modal SM signals respectively. Likewise, Nmon-Ti and Nmul-Ti indicates number of truly identified mono-and multimodal SM signals respectively. Furthermore, Nmon-Fi and Nmul-Fi are the number of SM signals which are falsely identified as mono-, and multi-modal SM signals respectively. Nto-Ti represents the total number of truly identified SM signals. In the last column, Rs represents the overall success rate of proposed techniques.

Fig. 6 .

 6 Fig. 6. Simulated mono-modal SM signals with SNR of 10 dB and 40 dB in case of (a) weak feedback regime (C = 0.1) and (b) moderate feedback regime (C = 4).

Table 1 .

 1 Values of statistical parameters of simulated normalized mono-modal SM signals for varying SNR under weak-feedback regime for C = 0.1, and amplitude = 5λ0.

	Techs.	Feats.	SNR (10 dB)	SNR (20 dB)	SNR (30 dB)	SNR (40 dB)
	VBT	varp-diff	0.015	0.012	0.012	0.008
	KBT	SMkur	1.718	1.452	1.433	1.429
	SBT	SMskw	0.149	0.144	0.142	0.141
	SKBT	||SMskur| 11.458	10.053	9.821	9.510
	Table 2. Statistical parameters' values for simulated
	normalized mono-modal SM signals for varying SNR under
	moderate-feedback regime (C = 4), and amplitude = 5λ0.
	Techs.	Feats.	SNR (10 dB)	SNR (20 dB)	SNR (30 dB)	SNR (40 dB)
	VBT	varp-diff	0.016	0.011	0.007	0.006
	KBT	SMkur	2.267	2.253	2.099	2.007
	SBT	SMskw	-0.345	-0.340	-0.326	-0.320
	SKBT |SMskur|	6.777	6.627	6.162	6.103

Table 3 .

 3 Extracted threshold values of proposed statistical

	parameters based on simulations on mono-modal SM
	signals under varying optical feedback, vibration
	amplitude, and signal to noise ratio
		VBT	KBT	SBT	SKBT
	Features	varp-diff	SMkur	SMskw	|SMskur|
	Threshold values	0.017	2.7	0.2	5

Table 3 .

 3 Results are presented in

Table 4

 4

Table 4

 4 Performance of proposed techniques by testing experimentally acquired dataset of 60 SM signals

	Techs.	Nmon /Nmul
	VBT	30/30	30	21	9	0	51	85
	KBT	30/30	24	21	9	6	45	75
	SBT	30/30	29	26	4	1	55	91
	SKBT 30/30	26	30	0	4	56	93
	MV	30/30	30	30	0	0		100

© 2020 Optical Society of America]. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibited


Joris Van Der Hoeven
email: vdhoeven@lix.polytechnique.fr

On the computation of limsups

Keywords: Asymptotic expansion, exp-log function, oscillating function, Diophantine approximation, algorithm A.M.S. subject classification: 03H05, 11J25, 11J61, 11J71, 11K06, 11K36, 12L15, 14P10, 14P15, 26A03, 26A09, 26A12, 26A48, 26E30, 28D99, 41A58, 41A60, 68Q25

In the last years, several asymptotic expansion algorithms have appeared, which have the property that they can deal with very general types of singularities, such as singularities arising in the study of algebraic differential equations. However, attention has been restricted so far to functions with strongly monotonic asymptotic behaviour: formally speaking, the functions lie in a common Hardy field, or, alternatively, they are determined by transseries.

In this article, we make a first step towards the treatment of functions involving oscillatory behaviour. More precisely, let be an algebraic function defined on [¡1; 1] q , let F 1 (x); : : : ; F q (x) be exp-log functions at infinity in x, and let (x) = (sin (F 1 (x)); : : : ; sin (F q (x))):

We give a method to compute limsup x!1 (x). Moreover, the techniques we use are stronger than this result might suggest, and we outline further applications.

Introduction

In the last years, several asymptotic expansion algorithms have appeared [START_REF] Shackell | Growth estimates for exp-log functions[END_REF][START_REF] Shackell | Limits of Liouvillian functions[END_REF][START_REF] Gonnet | Limit computation in computer algebra[END_REF][START_REF] Richardson | Expansions of exp-log functions[END_REF][START_REF] Van Der Hoeven | Automatic asymptotics[END_REF]. These algorithms are have the property that they can deal with very general types of singularities, such as singularities arising in the study of certain algebraic differential equations. However, attention has been restricted so far to functions with strongly monotonic asymptotic behaviour. This means that the functions lie in a common Hardy field, or, alternatively, that they are determined by transseries. In this article, we make a first step to the treatment of functions involving oscillatory behaviour. We also notice that Grigoriev obtained some very interesting related results in [START_REF] Grigoriev | Deviation theorems for solutions to differential equations and applications to lower bounds on parallel complexity of sigmoids[END_REF][START_REF] Grigoriev | Deviation theorems for solutions to linear ordinary differential equations and applications to lower bounds on parallel complexity of sigmoids[END_REF] although his more probabilistic point of view is different (even complementary) from ours.

The structure of this paper is as follows: in section 2, we recall a classical density theorem for linear curves on the n-dimensional torus (see for example [START_REF] Koksma | Diophantische approximationen[END_REF][START_REF] Kuipers | Uniform distribution sequences[END_REF]). In section 3, this theorem is generalized to more general classes of curves on the torus.

In section 4, we study exp-log functions at infinity: an exp-log function is a function which is built up from the rationals Q and x, using the field operations, exponentiation and logarithm. An exp-log function at infinity is an exp-log function which is defined in a neighbourhood of infinity. We present a more compact version of an expansion algorithm of exp-log functions at infinity, originally due to Shackell [START_REF] Shackell | Limits of Liouvillian functions[END_REF] (see also [START_REF] Richardson | Expansions of exp-log functions[END_REF]). For this, we assume the existence of an oracle for deciding whether an exp-log function is zero in a neighbourhood of infinity. This problem has been reduced to the corresponding problem for exp-log constants in [START_REF] Van Der Hoeven | Generic asymptotic expansions[END_REF][START_REF] Van Der Hoeven | Automatic asymptotics[END_REF]. A solution to the constant problem was given by Richardson in [START_REF] Richardson | How to recognise zero[END_REF], modulo Schanuel's conjecture: Conjecture 1. (Schanuel) If 1 ; : : : ; n are Q-linearly independent complex numbers, then the transcendence degree of Q[ 1 ; : : : ; n ; e 1 ; : : : ; e n ] over Q is at least n.

In section 5, we are given an algebraic function defined on [¡1; 1] q , and exp-log functions at infinity F 1 (x); : : : ; F q (x) in x. We show how to compute limsup x!1 (sin (F 1 (x)); : : : ; sin (F q (x))):

In section 5, we will assume the existence of an oracle for checking the Q-linear dependence of exp-log constants. Actually, Richardson's algorithm can easily be adapted to yield an algorithm for doing this modulo Schanuel's conjecture.

A density theorem on the n-dimensional torus

Let 1 ; : : : ; n be Q-linearly independent numbers: we will use vector notation, and denote the vector ( 1 ; : : : ; n ) by . In this section, we prove that the image of x 7 ! x, from R in the n-dimensional torus T n = R n /Z n is dense. Notice that we use the same notation for x and its class modulo Z n . Moreover, we show that the density of the image is uniform is a sense that will be made precise. The following theorem is classical: Theorem 2. (Kronecker) Let 1 ; : : : ; n be Q-linearly independent real numbers. Let e 1 ; : : : ; e n be the canonical base of R n . Then

1 e 1 Z + + n e n Z + R (e 1 + + e n ) is dense in R n .
Now let X be a measurable subset of T n , and let I be some interval of R. Denoting the Lebesgue measure by , we define We say that P holds for all I sufficiently close to infinity, if P holds for all sufficiently large a.

(I ; X) = (fx 2 I j x 2 X g) (I) : ( 1
The next theorem is also classical, but for convenience of the reader we present a proof, since similar techniques will be used in the next section: Theorem 4. (Bohr, Sierpi«ski, Weyl) Let 1 ; : : : ; n be Q-linearly independent real numbers and let be given by (1). Let

X = [a 1 ; b 1 [ [a n ; b n [ T n
be an n-dimensional block, with 0 6 a i 6 b i 6 1 for all i. Then lim

(I)!1 (I ; X) = (X); uniformly in I.
Proof. The theorem trivially holds if a i = 0 and b i = 1 for all but one 1 6 i 6 n. Hence, it suffices to prove the theorem when the a i 's and the b i 's are rational numbers. Indeed, let a 1 0 ; b 1 0 ;:::; a n 0 ; b n 0 be rational numbers with

ja 1 0 ¡ a 1 j + jb 1 0 ¡ b 1 j + + ja n 0 ¡ a n j + jb n 0 ¡ b n j 6 , and denote X 0 = [a 1 0 ; b 1 0 [ [a n 0 ; b n 0 [
. Then j(I ; X 0 ) ¡ (I ; X)j 6 2 for (I) sufficiently large, uniformly in I.

Because of Proposition 3(a) and (b), it suffices to prove the theorem for fixed p = (p 1 ;:::; p n ) 2 (N ) n and for all

X = X k = k 1 p 1 ; k 1 + 1 p 1 k n p n ; k n + 1 p n ;
with 0 6 k 1 < p 1 ; : : : ; 0 6 k n < p n . We remark that [0; 1[ n = `k X k , so that P k (I ; X k ) = 1. Now let " > 0. For each k, we can find x k , with d( x; k) < "/n, by Proposition 2. Consequently, we have (S ¡x k X k 4 X 0 ) < ", where A 4 B denotes the symmetric difference of A and B. Hence, (X l 4 S (x l ¡x k ) X k ) < 2 ", for each l with l 1 < p 1 ; : : : ; l n < p n . Using Proposition 3, we can now estimate

j(I ; X l ) ¡ (I ; X k )j 6 j(I ; S (x l ¡x k ) ) ¡ (I ; X k )j + (X l 4 S (x l ¡x k ) X k ) 6 j(S x k ¡x l I ; X k ) ¡ (I ; X k )j + 2 " 6 jx k ¡ x l j (I) + 2 ":
Taking (I) > jx k ¡ x l j/", for any k and l, we get

(I ; X k ) ¡ 1 p 1 p n 6 1 p 1 p n X k j(I ; X k ) ¡ (I ; X l )j < 3 ":
Hence j(I ; X k ) ¡ (X k )j < 3 ", for sufficiently large (I), uniformly in I. This completes our proof.

A more general density theorem

In this section we will obtain a more general uniform density theorem on the torus, when the application x 7 ! x from section 2 is replaced by a non linear mapping, which satisfies suitable regularity conditions. Before coming to this generalization, we will need some definitions and lemmas. We say that a function f defined in a neighbourhood of infinity is steadily dominated by x, if f has a continuous second derivative, f tends to infinity, f 0 decreases strictly towards zero, and f 00 / f 0 tends to zero. We remark that such functions admit functional inverses in a neighbourhood of infinity.

More generally, we say that if f and g are functions in a neighbourhood of infinity, such that g is invertible, then f is steadily dominated by g, if f g inv is steadily dominated by x. In this case, we write f s g. It is easily verified that if f s x and g s x, then f g s x, so that s is transitive. We also remark that if f s g and if h is a function, which has a continuous second derivative and tends to infinity, then f h s g h. We finally have the following property of steady domination: Lemma 5. Let h be steadily dominated by x and let l > 0 and " > 0 be given. Then for all sufficiently large x we have jh 0 (x + d) ¡ h 0 (x)j < " h 0 (x), for all d with jdj < l.

Proof. Let x 0 be such that jh 00 (x) / h 0 (x)j < " h 0 (x), for all x > x 0 ¡ l. We have jh 0 (x + d) ¡ h 0 (x)j 6 jd h 00 ()j < " h 0 (), for some between x and x + d. If d is positive, then h 0 () 6 h 0 (x), and we are done. In the other case, we have

jh 0 (x + d) ¡ h 0 (x)j 6 " h 0 (x) ¡ " jh 0 (x + d) ¡ h 0 (x)j, whence jh 0 (x + d) ¡ h 0 (x)j < ("/(1 ¡ ")) h 0 (x).
Now let X be a measurable subset of R. For each interval I, we define:

(I ; X) = (I \ X) (I) :
We say that X admits an asymptotic density (X) if lim

(I)!1 (I ; X) = (X);
uniformly in I, for I sufficiently close to infinity. More generally, if h is steadily dominated by x, then we say that X admits h-asymptotic density h (X) if lim

(h inv (I))!1 (I ; X) = (X);
uniformly in I, for I sufficiently close to infinity.

Lemma 6. Let X be a measurable subset of R and let h be steadily dominated by x. If (X) exists, then so does h (h(X)) and we have h (h(X)) = (X).

Proof. Let " > 0. Let l 2 R be such that j(I ; X) ¡ j < ", whenever (I) > l.

Let I = [; [ with (h inv (I)) > l and subdivide h inv (I) in q = b(h inv () ¡ h inv ())/lc > 1 parts of equal length l 0 > l: [(h inv (); h inv ())[ = [a 1 ; b 1 [ q q [a q ; b q [;
with b i = a i+1 for 1 6 i < q. Then we have

( ¡ ") X i=1 q Z a i b i h 0 (b i ) dx 6 X i=1 q Z a i b i X (x) h 0 (b i ) dx 6 (h inv (X) \ I) 6 X i=1 q Z a i b i X (x) h 0 (a i ) dx 6 ( + ") X i=1 q Z a i b i h 0 (a i ) dx:
By Lemma 5, for all sufficiently large x, we have jh 0 (x + d) ¡ h 0 (x)j 6 " h 0 (x), for all d with jdj 6 l 0 . Hence,

X i=1 q Z a i b i h 0 (x) dx ¡ X i=1 q Z a i b i h 0 (b i ) dx 6 X i=1 q Z a i b i jh 0 (x) ¡ h 0 (b i )j dx 6 " X i=1 q Z a i b i h 0 (x) dx = " (I);
and we have a similar estimation, when replacing b i by a i . Consequently,

( ¡ ") (1 ¡ ") (I) 6 (h(X) \ I) 6 ( + ") (1 + ") (I):
This completes our proof.

Let f 1 s s f p be continuous functions defined in a neighbourhood of infinity, which strictly increase towards infinity. Let i;j > 0 (1 6 j 6 n i ) be such that i;1 ; : : : ; i;n i are Q-linearly independent for each i. Now consider the curve g(x) = (f 1 ( 1;1 x); : : : ; f 1 ( 1;n 1 x); : : : ; f p ( p;1 x); : : : ; f p ( p;n p x)) on T n (n = n 1 + + n p ), which is defined for sufficiently large x. By analogy with the preceding section, we define

f ;g (I ; X) = (fx 2 I j g(f 1 inv (x)) 2 X g) (I) ; (2)
for intervals I sufficiently close to infinity, and measurable subsets X of T n .

Theorem 7. Let f 1 ; : : : ; f p ; g and be given as above and let

X = [a 1 ; b 1 [ [a n ; b n [ T n
be an n-dimensional block. Then

lim (I)!1 f ;g (I ; X) = (X);
uniformly, for intervals sufficiently close to infinity.

Proof. We proceed by induction over p. If p = 0, we have nothing to prove. As before, it suffices to prove the theorem for multidimensional blocks X = X 1 X ~, with X 1 T n 1 and X ~ T n ~, where n ~= n 2 + + n p . We denote by g 1 (x) resp. g ~(x) the projections of g(x) on T n 1 resp. T n ~, when considering T n as the product of T n 1 and T n ~. Without loss of generality, we may assume that f 1 = x. Given a subset A of R or T n and its frontier @A, we denote for any " > 0

" A = fx 2 A j d(x; @A) > "g: Let " > 0. If g 1 (x) 2 " X 1 , then g 1 (x + d) 2
X 1 for all d with jdj < l, where l = max (1/ 1;1 ;:::; 1/ 1;n 1 ) ". Hence, for I sufficiently close to infinity,

I \ g 1 inv ( " X 1 ) (I \ l g 1 inv (X 1 )) + ]¡l; l[ I \ g 1 inv (X 1 ):
Therefore, Theorem 4 implies that for I sufficiently close to infinity

(I \ g 1 inv (X 1 )) (I) ¡ (X 1 ) < " (3) and (using that ( " X 1 4 X 1 ) < 2 n 1 ) ((I \ l g 1 inv (X 1 )) + ]¡l; l[) (I) ¡ (I \ g 1 inv (X 1 )) (I) 6 (2 n 1 + 1) ": (4) Now (I \ l g 1 inv (X 1 )) + ]¡l; l[ is a finite union of intervals, say I \ l g 1 inv (X 1 ) \ ]¡l; l[ = I 0 q q I q+1 ;
where I 1 ; : : : ; I q have length at least 2 l, and where I 0 and I q+1 have length at most 2 l. By the induction hypothesis, we have lim

(J)!1 (J \ f 2 (g ~inv (X ~))) (J) = (X ~);
uniformly, for J sufficiently close to infinity. Using Lemma 6 for h = f 2 inv , this gives us lim

(f 2 (J))!1 (J \ g ~inv (X ~)) (J) = (X ~);
uniformly, for J sufficiently close to infinity. In particular, we have

(J \ g ~inv (X ~)) (J) ¡ (X ~) < ";
for all J sufficiently close to infinity, with (J) > l. Thus, choosing I sufficiently close to infinity, we have

(I i \ g ~inv (X ~)) (I i ) ¡ (X ~) < ";
for all 1 6 i 6 q.

Taking (I) > 2 l /", and using (3) and (4), this gives us

j f ;g (I ; X) ¡ (X)j 6 ( `i=0 q+1 I i \ g ~inv (X ~)) (I) ¡ (X) + (I \ g inv (X)) (I) ¡ ( `i=0 q+1 I i \ g ~inv (X ~)) (I) 6 X i=0 q+1 (I i \ g ~inv (X ~)) ¡ (X ~) (I i ) (I) + (X ~) ( `i=0 q+1 I i ) (I) ¡ (X) + (2 n 1 + 1) " 6 X i=1 q (I i ) " (I) + (4 n 1 + 5) " 6 (4 n 1 + 6) ":
This completes the proof.

Expansions of exp-log functions at infinity

Let T denote the field of germs at infinity of exp-log functions and C the subfield of explog constants. Elements of T can be represented by exp-log expressions i.e. finite trees whose internal nodes are labeled by +; ¡; ; /; exp or log, and whose leaves are labeled by x or rational numbers. The set of exp-log expressions which can be evaluated in a neighbourhood of infinity is denoted by T expr . We have a natural projection f 7 ! f from T expr onto T. We make the assumption that we have at our disposal an oracle which can decide whether a given exp-log expression in T expr is zero in a neighbourhood of infinity.

In view of [START_REF] Van Der Hoeven | Generic asymptotic expansions[END_REF][START_REF] Van Der Hoeven | Automatic asymptotics[END_REF] it actually suffices to assume the existence of an oracle to decide whether a given exp-log constant is zero.

Grid-based series

Let us first recall some basic concepts. An effective asymptotic basis is an ordered finite set fb 1 ; : : : ; b n g of positive infinitesimal exp-log expressions in T expr , such that log b i log b i+1 (i.e. log b i = o(log b i+1 )) for 1 6 i 6 n ¡ 1. For instance, the set B = flog ¡1 x; x ¡1 ; e ¡x 2 g is an effective asymptotic basis. An effective asymptotic basis B generates an effective asymptotic scale, namely the set S B of all products b 1 1 b n n of powers of the b i 's, with the i 's in C. Elements of S B are also called monomials.

Given an effective asymptotic basis B, let G B expr denote the set of expressions which are built up from C; S B ; +; ¡; ; / and the operations " 7 ! exp ", resp. " 7 ! log (1 + "), for infinitesimal ". We observe that f can be expanded as a series in b n with coefficients in G fb 1 ; : : : ;b n¡1 g expr . Moreover, these coefficients can recursively be expanded in b n¡1 ; : : : ; b 1 :

f = X n 2C f n b n n f n ; : : : ; 2 = X 1 2C
f n ; : : : ; 1 b 1 1 :

The exp-log expressions of the form f n ; : : : ; i are called iterated coefficients of f . In particular, the iterated coefficients of the form f n ; : : : ; 1 are exp-log constants.

The above expansions of f have an important property [START_REF] Van Der Hoeven | Automatic asymptotics[END_REF]: the support of f as a series in b n is included in a set of the form 1 N + + p N + , where the i 's and are constants in C we say that f is a grid-based series. From this property, it follows that the support of f is well-ordered.

Another important property of the expansion of f in b n and the expansions of its iterated coefficients is that they can be computed automatically. By this we mean that for each integer i, we can compute the first i terms of the expansion of f and so can we for its iterated coefficients. In particular, we can compute the sign of f , test whether f is infinitesimal, etc.

For the computation of the expansions of f in b n , we use the usual Taylor series formulas. In the case of division 1/ f , we compute the first term f b n of f and then use the formula

1/ f = (1/ f ) b n ¡ (1/(1 + ")), where " = (f / f b n ) ¡ 1.
The only problem when applying these formulas is that we have to avoid indefinite cancelation: note that indefinite cancelation only occurs if after having computed the first i terms of the expansion, f is actually equal to the sum of these terms. But this can be tested using the oracle, and we stop the expansion in this case.

Automatic expansions of exp-log expressions

The asymptotic expansion algorithm takes an exp-log expression f 2 T expr on input, computes a suitable effective asymptotic basis B and rewrites f into an element of G B expr . The main idea of the algorithm lies in imposing some suitable conditions on B: we say that a linearly ordered set B = fb 1 ; : : : ; b n g is an effective normal basis if NB1. B is an effective asymptotic basis. Such a basis is constructed gradually during the algorithm i.e. B is a global variable in which we insert new elements during the execution of the algorithm, while maintaining the property that B is an effective normal basis. We also say that B is a dynamic effective normal basis. Let us now explicitly give the algorithm, using a PASCAL-like notation:

NB2. log b i 2 G fb

Algorithm expand(f ). The algorithm takes an exp-log expression f 2 T expr on input and rewrites it into a grid-based series in G B expr , where the global variable B contains an effective normal basis which is initialized by B := fx ¡1 g.

case f 2 Q: return f case f = x: return (x ¡1 ) ¡1 case f = g > h,
g ¡ := g ¡ g + B := B [ fe ¡jg + j g return (e ¡jg + j ) ¡signg + e g ¡
Let us comment the algorithm. The first three cases do not need explanation. In the case f = log g, the fact that B is an effective normal basis is used at the end:

1 log b 1 + + n log b n is indeed an expression in G B expr
. The expansion of the exponential of a bounded series g is done by a straightforward Taylor series expansion. If g is unbounded, then we test whether g is asymptotic to the logarithm of an element in B i.e. we test whether := lim g /log b i is a non zero finite number for some i. If this is so, then f = b i e g ¡logb i and e g ¡logb i is expanded recursively. We remark that no infinite loop can arise from this, because successive values of g in such a loop would be asymptotic to the logarithms of smaller and smaller elements of B, while B remains unchanged. Finally, if g is not asymptotic to the logarithm of an element in B, then B has to be extended with an element of the order of growth of f . The decomposition g = g + + g ¡ is computed in order to ensure that B remains an effective normal basis.

On the automatic computation of limsups

In this section we show how Theorem 7 can be applied to compute limsups (or liminfs) of certain bounded functions, involving trigonometric functions. The idea is based on the following consequence of Theorem 7.

Theorem 8. Let 1 f 1 f p be exp-log functions at infinity. Let i;j > 0 ( 1 6 j 6 n i ) be such that i;1 ;:::; i;n i are Q-linearly independent for each i. Denote U = fx + ¡1 p y 2 Cj x 2 + y 2 = 1g and n = n 1 + + n p . Let be a continuous function from U n into R and let

(x) = (e ¡1 p 1;1 f 1 (x)
; : : : ; e

¡1 p p;np f p (x)
):

Then limsup x!1 (x) = sup x2U n (x):
Proof. We first notice that we will be able to apply Theorem 7 on our input data: by a well known theorem, which goes back to Hardy [START_REF] Hardy | Properties of logarithmico-exponential functions[END_REF], the germs at infinity of f 1 ; : : : ; f p lie in a common Hardy field. Consequently, f 1 s s f p , and f 1 ; : : : ; f p are strictly increasing in a suitable neighbourhood of infinity. The mapping is defined in a neighbourhood V of infinity, and can be factored where 1 and 2 are both continuous. Since T n is compact, there exists a point x in which 2 attains its maximum. Let " > 0. There exists a neighbourhood V of x, such that j 2 (y) ¡ 2 (x)j < ", for any y in V . By Theorem 7, there exist x, with 1 (x) 2 V as close to infinity as we wish. For such x, we have j (x) ¡ sup x2U n (x)j < ".

V ! ! ! ! ! ! ! ! ! ! R = V ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! 1 T n ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! 2 R, with 1 (x) = 1;1 f 1 (x) 2
We now turn to the computation of this limit.

Theorem 9. Let F 1 ; : : : ; F q be exp-log functions at infinity. Let : U q ! R a real algebraic function, where we consider U q as a real algebraic variety. Assume that we have an oracle to test the Q-linear dependence of exp-log constants. Then there exists an algorithm to compute the limsup of (x) = (e ¡1 p F 1 (x) ; : : : ; e ¡1 p F q (x) ).

Proof. Using the identity e ¡x = 1/e x , we may always assume without loss of generality, that the F i 's are all positive. Now the algorithm consists of the following steps:

Step 1. Compute a common effective normal basis for F 1 ;:::; F p , using the algorithm from section 4. Order the F i 's w.r.t. ; that is, F i F j or F i F j , whenever i < j.

Step 2. Simultaneously modify the F i 's and the algebraic function in the e ¡1 p F i 's, until we either have F i F j , or F i = F j , for some , whenever i < j. As long as this is not the case, we take j maximal, such that the above does not hold, and do the following:

First compute the limit of Step 3. Compute exp-log functions f 1 f p , and constants i;j (1 6 j 6 n i ), such that each F l can be written as F l = i;j f i , for some i and j. Replacing e ¡1 p F i by its limit for each bounded F i , we reduce the general case to the case when 1 f 1 .

F i /F j . Next insert F i 0 := F i ¡ F j and F j 0 := F j into
Step 4. This step consists in making the i;j 's Q-linearly independent for each fixed i. Whenever there exists a non trivial Q-linear relation between the i;j 's (for fixed i), we may assume without loss of generality that this relation is given by

a n i i;n i = a 1 i;1 + + a n i ¡1 i;n i ¡1 ;
for a 1 ;: : : ; a n i in Z and a n i > 0. As long as we can find such a relation, we do the following: For all j < n i , replace i;j by i;j 0 := i;j / a n i and e ¡1 p i; j f i by (e ¡1 p i; j ) a n i ¡1 in the expression for i.

0 f i ) a n i in
Step 5. By Theorem 8, the limsup of is the maximum of on U n , where n = n 1 + + n p . To compute this maximum, we determine the set of zeros of the gradient of on U n . Then is constant on each connected component and the maximum of these constant values yields max U n . To compute the zero set of the gradient of and its connected components, one may for instance use cylindrical decomposition (see [START_REF] Collins | Quantifier elimination for real closed fields by cylindrical algebraic decomposition[END_REF]). Of course, other algorithms from effective real algebraic geometry can be used instead.

The correctness of our algorithm is clear. The termination of the loop in step 2 follows from the fact that the new F i 0 is asymptotically smaller then F j , so that either the -class of F j strictly decreases, or the number of i's with F i F j , but not F i = F j for some . The number of -classes which can be attained is bounded by the initial value of q.

Corollary 10. Let F 1 ;:::; F q be exp-log functions at infinity and be an algebraic function in q variables, defined on [¡1; 1] q . Assume that we have an oracle to test the Q-linear dependence of exp-log constants. Then there exists an algorithm to compute the limsup of (x) = (sin (F 1 (x)); : : : ; sin (F q (x))).

Example 11. Consider the function (x) = 2 sin x 2 ¡ sin (x 3 /(x ¡ 1)) 3 + sin e x 2 ¡ sin (e x 2 + 1) :

The first step consists in expanding x 2 = x 2 , x 3 /(x ¡ 1) = x 2 + x + , e x 2 = e x 2 and e x 2 + 1 = e x 2 + 1. All these functions have the same -class, but they are not all homothetic. Therefore, some rewriting needs to be done. First, ; which corresponds to the rewriting sin (e x 2 + 1) = sin e x 2 cos 1 + sin 1 cos e x 2 :

x 3 /(x ¡ 1) = x 2 + x 2 /(x ¡
In step 4, no Q-linear relations are found, so that we have to determine the maximal value of

Conclusion

We have shown how to compute limsups of certain functions involving trigonometric functions, exponentiation and logarithm. Actually, the techniques we have used are far more general than Theorem 9 might suggest. Let us now briefly mention some generalizations.

For more details, we refer to [START_REF] Van Der Hoeven | Automatic asymptotics[END_REF].

In Theorem 9, the crucial property of the functions F 1 ; : : : ; F q is that they are strongly monotonic and that we have an asymptotic expansion algorithm for them. Consequently, more general functions than exp-log functions can be taken instead, like Liouvillian functions, functions which are determined by systems of real exp-log equations in several variables, etc.

The crucial property of the function is that it belongs to a class for which a cylindrical decomposition algorithm exists. Again, more general classes of functions can be considered. In particular, modulo suitable oracles, one can consider the class of solutions to real explog systems in several variables.

Our techniques can also be used to compute automatic asymptotic expansions of sinexp-log functions at infinity of trigonometric depth one (i.e. without nested sines). However, some difficult number theoretical phenomena may occur in this case, as the following example illustrates:

2 ¡ sin x ¡ sin e x > 1 1 ¡ (x + 2) :
This asymptotic inequality follows from the number theoretical properties of e. In general, such inequalities are very hard to obtain (if decidable at all!): a systematic way to obtain them would in particular yield solutions to deep unsolved problems in the field of Diophantine approximation (for a nice survey, see [START_REF] Lang | Transcendental numbers and diophantine approximation[END_REF]). Nevertheless, we notice that the above example is degenerate in the sense that 2 is precisely equal to the limsup of sin x + sin x 2 . In the generic case, a complete asymptotic expansion for sin-exp-log functions at infinity of trigonometric depth one does exist. In the degenerate case, we need assume the existence of a suitable oracle for Diophantine questions.

)

  Let us also denote by d the Euclidean distance on T n . Let S d , resp. S d denote the shift operator on R (resp. R n or T n ): S d (x) = x + d and S d (x) = S (d 1 ; : : : ;d n ) (x 1 ;:::; x n ) = (x 1 + d 1 ;:::; x n + d n ) = x + d. The following are immediate consequences of the definition of : Proposition 3. We have a) (I ; X) = P i2N (I ; X i ), if X = `i2N X i . b) j(I ; X) ¡ (S d I ; X)j 6 jdj/ (I), for all d. c) (I ; X) = (S ¡d I ; S d X), for all d. It will be convenient to adopt some conventions for intervals I = [a; b] (resp. I = [a; b[; I = ]a; b] or I = ]a; b[) whose lengths b ¡ a tend to infinity: we say that a property P holds uniformly in I, if the property holds uniformly in a: 9l 0 ; 8a; 8l > l 0 ; P ([a; a + l]):

  1 ; : : : ;b i g expr for all i > 1, where log log b i ¡1 log b i . NB+. b 1 = log l ¡1 x for some l 2 N, where log l x = def log : : : l times log.

  where >2f+; ¡; ; /g: if > = / and h = 0 then error division by zero return expand(g) > expand(h) case f = log g: g := expand(g) Denote B = fb 1 = log l ¡1 x; b 2 ; : : : ; b n g. if g 6 0 then error invalid logarithm Rewrite g = c b 1 1 b n n (1 + "), with infinitesimal " in G B expr and c 2 C. if 1 = / 0 then B := B [ flog l+1 ¡1 xg return log c + 1 log b 1 + + n log b n + log (1 + ") case f = e g : g := expand(g) Denote B = fb 1 ; : : : ; b n g. if g = O(1) then return e c e g ¡c , where c := g 91<i6n; g log b i then := lim g /log b i return b iexpand(e g¡logb i ) Let i be such that log jgj log b i . g + := g

Figure 1 .

 1 Figure 1. Plot of the function from Example 11.

  the set of F i 's and remove F i . The new expression for is obtained by replacing each e ¡1

				p	F i
	by e ¡1 p	F i 0	e ¡1 p	F j 0 .

Joris van der Hoeven

On the computation of limsups


Chloé Blaise
email: blaisechloe@gmail.com

Christophe Mazzia
email: christophe.mazzia@univ-avignon.fr

Armin Bischoff
email: armin.bischoff@univ-avignon.fr

Alexandre Millon
email: alexandre.millon@imbe.fr

Philippe Ponel

Olivier Blight
email: olivier.blight@univ-avignon.fr

The key role of inter-row vegetation and ants on predation in Mediterranean organic vineyards

Keywords:

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

A standard practice in conventional viticulture is the removal of inter-row vegetation using herbicides to reduce weed competition [START_REF] Winter | Effects of vegetation management intensity on biodiversity and ecosystem services in vineyards: a meta-analysis[END_REF]. However, both farmers and society at large are increasingly aware of the need to move from high-input agriculture towards more environmentally sound and sustainable farming practices. Glyphosate, the most widely used herbicide in agriculture [START_REF] Benbrook | Trends in glyphosate herbicide use in the United States and globally[END_REF], represents an emblematic example of the need to change a chemistry-based agricultural paradigm dominant since the 1950s [START_REF] Pimentel | Green revolution agriculture and chemical hazards[END_REF]. In 2017, over 1.3 million Europeans signed a petition calling for a ban on this herbicide. Concern over human and environmental exposure to pesticides has led several European countries such as Germany, France, Italy, Austria and Greece to plan a complete ban on Glyphosate in the coming years. This growing awareness of the need for successful ecological change in agriculture offers an opportunity to rethink vegetation management and to move towards more sustainable and biodiversity-friendly farming practices. This is particularly relevant in vineyards, a perennial crop system with a high pesticide consumption [START_REF] Butault | L'utilisation Des Pesticides En France: Etat Des Lieux Et Perspectives De Réduction[END_REF].

More sustainable vineyard practices like wider use of vegetated inter-rows remain limited by various constraints, such as potential competition for water, particularly in water-limited regions of the Mediterranean [START_REF] Celette | Competition for nitrogen in an unfertilized intercropping system: the case of an association of grapevine and grass cover in a Mediterranean climate[END_REF][START_REF] Ruiz-Colmenero | Soil and water conservation dilemmas associated with the use of green cover in steep vineyards[END_REF]. However, vegetation cover can directly and indirectly provide several ecosystem services to wine and grape production when managed appropriately [START_REF] Garcia | Management of service crops for the provision of ecosystem services in vineyards: a review[END_REF][START_REF] Winter | Effects of vegetation management intensity on biodiversity and ecosystem services in vineyards: a meta-analysis[END_REF]. Vegetation cover in vineyards can enhance biological activity [START_REF] Quecedo | Soil quality evaluation following the implementation of permanent cover crops in semi-arid vineyards. Organic matter, physical and biological soil properties[END_REF]

Materials and methods

Study sites and design

This study was conducted in south-eastern France, between the southern slopes of the Luberon mountains and the Durance river. The landscape is dominated by medium-sized vineyards (0.7 +/-0.5 ha) and the degree of urbanisation is low. In this region, winegrowers most often keep bare soil from mid-spring to autumn rains. This is the most widely adopted practice. In permanently vegetated vineyards, the vegetation is mowed or laid down before summer. The southern Luberon area has a Mediterranean climate, with hot and dry summers followed by mild winters and irregular precipitation in autumn and/or spring. Mean annual temperatures and total annual precipitation for the period of 2003-2018 are 13.5 • C and 676 mm, respectively.

We selected 26 vineyards involving three different inter-row management types : (i) tillage (0/2) (N = 8), (ii) partially vegetated (one out of two inter-rows vegetated) (1/2) (N = 10) and (iii) all inter-rows vegetated (2/2) (N = 8). All selected vineyards were under organic management, with an average size of 7 008 ± 746 m 2 and scattered across an area of 20 km by 6 km (Fig. 1). According to the French guidelines for organic viticulture no chemical was applied to the 26 vineyards for at least three years. The inter-rows of all these vineyards were sown at least once in the last five years with commercial grasslegume mixtures. However, with the exception of two vineyards, they were all dominated by spontaneous species (Appendix A). All were planted with either Syrah or Grenache grapevines, the most common varieties in this region.

The lepidoptera Lobesia botrana (Lepidoptera: Tortricidae) and the leafhopper Scaphoideus titanus (Hemiptera: Cicadellidae) are the main pest insects in the Luberon. The moth L. botrana feeds on grapes, reducing yield and increasing susceptibility to fungal infections. Lobesia botrana is among the most economically important insects in Europe. Scaphoideus titanus is the vector of a phytoplasma that causes the infection by Flavescence dorée. Flavescence dorée causes yield losses and lower grape quality [START_REF] Chuche | Biology and ecology of the Flavescence dorée vector Scaphoideus titanus: a review[END_REF]. Scaphoideus titanus completes one reproductive cycle in spring and L. botrana up to four from early spring to late summer. Lobesia botrana larvae can reach a length of 8-9 mm, compared to 3-4 mm for larvae of S. titanus.

Characterisation of vegetation cover and arthropod community

A pair of inter-rows separated by two inter-rows was selected in the centre of each vineyard (Fig. 2). The plant community was recorded in three 2 × 2 m quadrats in each selected inter-row from 13 to 22 May 2019, resulting in a total of six quadrats per vineyard. We identified vascular plant species and estimated their individual percentage cover as well as the total percentage of vegetation cover in each quadrat. Plant percentage cover was estimated as the vertical projection of all aboveground organs. Data were averaged across quadrats within each vineyard.

In order to estimate the most abundant and active predators (Formicidae, Arachnids [spiders, harvestmen] and Carabidae), pitfall traps were placed in two inter-rows in each vineyard and removed after one week. Traps (clear plastic) were 11 cm deep and 8 cm in diameter. They were buried to the rim and filled with propylene glycol to a quarter of the depth. Two pitfall traps were placed 20 m apart in the middle of each of the two inter-rows, resulting in a total of 104 pitfall traps. To avoid an influence of predator trapping on the sentinel prey experiment, the pitfall traps were only placed after the end this experiment in July.

Predatory arthropods were sorted and identified to order for arachnids (Araneae and Opiliones) and to family for Hymenoptera (Formicidae) and Coleoptera (Carabidae) in the laboratory, counted and stored in a 70 % ethanol solution. Our focus being on generalist predators able to prey on larvae at last instar stage, only ground beetles and arachnids larger than 5 mm were considered. [START_REF] Buchholz | Soil biota in vineyards are more influenced by plants and soil quality than by tillage intensity or the surrounding landscape[END_REF], water infiltration [START_REF] García-Díaz | Nitrogen losses in vineyards under different types of soil groundcover. A field runoff simulator approach in central Spain[END_REF] and organic matter availability [START_REF] Ruiz-Colmenero | Soil and water conservation dilemmas associated with the use of green cover in steep vineyards[END_REF]. Maintaining vegetation cover improves soil stabilisation because organic matter stabilises aggregates, and root systems protect the soil [START_REF] Ruiz-Colmenero | Soil and water conservation dilemmas associated with the use of green cover in steep vineyards[END_REF]. Vegetation cover can also favour beneficial organisms associated with grapevines [START_REF] Altieri | Vegetation management and biological control in agroecosystems[END_REF]Sáenz-Romo et al., 2019b) and supports higher levels of biodiversity [START_REF] Altieri | Manipulating vineyard biodiversity for improved insect pest management: case studies from northern California[END_REF]Wilson et al., 2017b).

Pest regulation has been identified as another important ecosystem service enhanced by the presence of inter-row vegetation. The pest regulation service has commonly been linked to the abundance of natural enemies, such as generalist predators and parasitoids [START_REF] Blaise | Abundance of spiders and insect predators on grapes in central California[END_REF][START_REF] Sharley | The effects of soil tillage on beneficial invertebrates within the vineyard[END_REF][START_REF] Hoffmann | Can Flowering Greencover Crops Promote Biological Control in German Vineyards?[END_REF]Sáenz-Romo et al., 2019b). Predators are typically larger than their prey and, unlike parasitoids, they require more than one prey individual during their development. Generalist predators that feed on a variety of prey, such as leafhoppers, flies and caterpillars, are commonly considered to be poorly effective in pest regulation because their numbers are not sufficient to deal with pest population dynamics [START_REF] Miñarro | Colonization of apple orchards by predators of Dysaphis plantaginea: sequential arrival, response to prey abundance and consequences for biological control[END_REF]. However, generalist predator efficiency may be higher if their abundances are high before the pest invades [START_REF] Symondson | Can generalist predators be effective biocontrol agents?[END_REF]. In the case of such early arrival, they may hamper or even prevent pest invasion.

In vineyards, spiders and predatory beetles, such as Carabidae and Staphylinidae, are among the largest generalist predator groups feeding on pests [START_REF] Thomson | Spatial scale of benefits from adjacent woody vegetation on natural enemies within vineyards[END_REF][START_REF] Pfingstmann | Contrasting effects of tillage and Agriculture, Ecosystems and Environment 311[END_REF]Sáenz-Romo et al., 2019b). To a lesser degree, lacewings, hoverflies and thrips have been recorded as vineyard predators [START_REF] Blaise | Abundance of spiders and insect predators on grapes in central California[END_REF][START_REF] Hoffmann | Can Flowering Greencover Crops Promote Biological Control in German Vineyards?[END_REF][START_REF] Loni | The Genus Paragus latreille (Diptera Syrphidae), a Useful Bioindicator in the Vineyards[END_REF]. Although ants are known to favour aphids and coccids through mutualistic interactions (exchanging honeydew for protection, [START_REF] Mansour | Interactions between the ant Tapinoma nigerrimum (Hymenoptera: formicidae) and the main natural enemies of the vine and citrus mealybugs (Hemiptera: pseudococcidae)[END_REF]; Beltra ` et al. ( 2017)), they may also act as predators of insect pests [START_REF] Offenberg | Ants as tools in sustainable agriculture[END_REF]. All these taxa play a significant role in crop pest regulation [START_REF] Symondson | Can generalist predators be effective biocontrol agents?[END_REF] and have been shown to be promoted by vegetation cover in vineyards [START_REF] Thomson | Vegetation increases the abundance of natural enemies in vineyards[END_REF]Sáenz-Romo et al., 2019a). Vegetation may also facilitate grapevine access to ground-dwelling predators [START_REF] Frank | Video analysis to[END_REF][START_REF] Irvin | The effect of an irrigated cover crop on grape vine productivity, and beneficial insect and grape pest abundance in southern California[END_REF].

In general, the abundance of natural enemies is higher in more diverse ecosystems because vegetation cover provides both additional habitat and food [START_REF] Nicholls | Reducing the abundance of leafhoppers and thrips in a northern California organic vineyard through maintenance of full season floral diversity with summer cover crops[END_REF]. However, there have been few attempts to document how vegetation cover enhances predation. The relationship between predation and vegetation cover has been explored in several recent studies, but with contrasting results. While [START_REF] Rusch | Grape moth density in Bordeaux vineyards depends on local habitat management despite effects of landscape heterogeneity on their biological control[END_REF] did not record any direct influence of local vegetation management on pest predation, both native and introduced vegetation cover were found to increase sentinel egg predation in other studies [START_REF] Danne | Effects of native grass cover crops on beneficial and pest invertebrates in Australian vineyards[END_REF][START_REF] Hoffmann | Can Flowering Greencover Crops Promote Biological Control in German Vineyards?[END_REF]. Further investigations are required to improve our understanding of vegetation management effects on both generalist predator abundance and their predatory activity.

Although vegetation cover may help increase generalist predator abundances, predators must move from the inter-row to the grapevine to affect pest insect control. In the present study, we evaluated the abundance of generalist arthropod predators and the related predation function in Mediterranean organic vineyards under three different interrow management systems. Our predation survey was performed both throughout the day and at night's end, to analyse the effects of vegetation cover, plant species richness and diversity on predators and on the attack on sentinel prey. We addressed the following research questions:

(1) Do generalist predator abundance and final predation increase with vegetation cover? (2) Do the effects on predator numbers and final predation depend on vegetation characteristics (plant species richness, percentage of vegetation cover and its variance, beta diversity (Bray-Curtis index)? (3) Are these effects limited to predation in the inter-row, or is there a spill-over effect to grapevine plants? (4) What are the major predator groups attacking sentinel prey?

Sentinel prey experiment and predator observations

Larvae of Lucilia sp. (Diptera, Calliphoridae) were used as sentinel prey to characterise predation by arthropod generalist predators. Sentinel prey can provide a direct, quantitative measure of predation under field conditions. Live sentinel prey provides more realistic data compare to artificial prey [START_REF] Lövei | A review of the sentinel prey method as a way of quantifying invertebrate predation under field conditions[END_REF]. All larvae used for the experiment were at last instar stage, sized ca. one centimetre and white. The sentinel preys were pinned on white cards (2 × 2 cm) for better detection. Forty-eight sentinel cards were placed in each vineyard: 24 per transect, 5 m apart, 12 placed directly on the ground in the middle of the inter-row and 12 attached to the vine trunk about 60 cm high. Cards were placed in the two transects used for vegetation and arthropods sampling. The experiment was conducted over 17 days in June 2019. Observations began at 10 am. The cards were checked four times, after 1, 2, 4.5 and 7 h, and finally removed at 8 pm. New cards were placed in the same position from 9 pm to 8.30 am to assess the level of predation occurring at night. During each survey, we classified the maggots under "presence", "absence" and "attacked". Arthropods eating sentinel preys were visually identified when present. A predation event was recorded if at least one maggot was missing or showed clear signs of attack. Most unattacked larvae remained alive at the end of both exposure sessions.

Data analysis

Mixed models were computed to explore the effects of management type and vegetation characteristics on different response variables (predator abundance per pitfall trap, final predation (card level), dynamics of predation (survival time) and observations of identified predators (card level)). The vineyard was systematically used as a random explanatory variable. Models were either with management or with vegetation variables. Although vegetation cover and inter-row management are highly correlated, these two approaches are complementary and thus straightforward. Models testing the effect of the interrow management provide information on impact of agricultural practices and models testing the effect of vegetation variables provide information on biotic interactions between predators and vegetation.

We performed a principal component analysis (PCA) based on (i) mean percentage of vegetation cover and (ii) species richness per quadrat within each vineyard, (iii) within vineyard variance of percentage of vegetation cover across the six quadrats, and (iv) within vineyard beta diversity calculated using the Bray-Curtis dissimilarity index for all pairs of quadrats within vineyards, to reduce the number of vegetation cover variables and to avoid correlation among these variables. We extracted the scores of the first two principal components, PC1 and PC2, which accounted for 45.6 % and 41.2 % of total variance, respectively. The first axis was related to species richness and percentage of vegetation cover, and the second axis was related to beta diversity and variance of percentage of vegetation cover. To facilitate interpretation of the interaction, PC1 and PC2 were considered binary variables. Appendix B), with nearly 50 % more ants in 2/2 (mean ± se = 47.5 ± 7.92) vineyards than in 0/2 (mean ± se = 24.9 ± 5.49) (P = 0.03). Higher variance in percentage of vegetation cover and beta diversity (PC2) were associated with higher abundances of ants (Table 2). Abundances of both spiders and harvestmen depended on the interaction between PC1 (percentage of vegetation cover and plant species richness) and PC2 (Table 2). When beta diversity and vegetation cover variance were high, spiders were significantly more abundant under a higher percentage of vegetation cover. When beta diversity and vegetation cover variance were low, spiders were more abundant under a lower percentage of vegetation cover.

Sentinel prey experiment

The probability of larvae being attacked during daytime was significantly affected by inter-row management type, card position, and their interaction (Table 2). Sentinel prey was significantly less attacked in 0/2 vineyards than in 1/2 vineyards when cards were on the ground, and significantly less than in 2/2 vineyards when cards were on the vine trunk (P < 0.004 and P = 0.02 respectively). Larvae located on the ground were significantly more attacked than those on the vine trunk (P = 0.01, P < 0.001 and P < 0.001, respectively for 0/2, 1/2 and 2/2) PCA scores below the mean of all vineyards were transformed to 0 and PCA scores above the mean were transformed to 1. We first modelled the response of predator abundance in pitfall traps to inter-row management. We conducted these analyses for each predator taxon (ants, spiders, carabids, harvestmen).

Regarding the sentinel prey experiment, a predation event was counted when at least one maggot was missing or showed clear signs of attack. We included the vineyard as a random effect in all analyses of sentinel prey.

Final predation was divided into daytime and night-time events and analysed separately. We used the number of predated larvae as the statistical unit (i.e. the number of larvae attacked or missing from the 48 cards placed in each vineyard). Differences in levels of both diurnal and nocturnal predation according to type of inter-row management were assessed using a GLMM with a binomial distribution and logit link. The card position (ground vs. vine trunk) was added as a fixed explanatory variable. Models included the interaction between inter-row management type and card position. To obtain a better understanding of how the inter-row vegetation parameters affected both diurnal and nocturnal final predation, we applied two separate GLMMs. Both models were run with PC1, PC2 and card position (ground or vine trunk) as explanatory variables and all their possible interactions.

A [START_REF] Kaplan | Nonparametric estimation from incomplete observations[END_REF] survival analysis was performed to calculate survival curves, and curves were plotted using the ggsurvplot function of the survminer package. Cox proportional hazard model survival analyses were used to obtain a temporal view of the overall diurnal predation process across the three inter-row management types. The first model related the survival times of sentinel prey to inter-row management type and card position (ground or vine trunk), as well as their interaction, as fixed explanatory variables. The second model related survival times to PC1 and PC2, card position and their interactions. The models were specified using the "coxme" function in R package coxme. Post-hoc tests were run using the emmeans function (emmeans package) to compare the three management types and the two sentinel card positions in cases of significant predictor effects. Finally, we separately explored the effects of inter-row management type and of PC1 and PC2 in interaction with card position on the abundance of individual predator species identified on sentinel prey, using GLMMs with a binomial distribution.

A stepwise selection was employed for model selection using the anova function. Data analyses were performed using RStudio, version 1.2.5033. Packages used here: MASS, glmmTMB, emmeans, survfit, coxme.

Results

Vegetation cover

We recorded 91 plant species in the 0/2 vineyards, predominantly annuals. Lolium rigidum (annual), Convolvulus arvensis (perennial) and Avena sterilis (annual) were the dominant species in the quadrats, with a mean cover of 5.4 %, 1.8 % and 1.7 %, respectively. In the 1/2 vineyards, 115 plant species were sampled, with the dominant species Lolium rigidum, Bromus hordeaceus and Vicia sativa covering 9.2 %, 3.5 % and 3.3 % of the quadrats, respectively. We identified 114 plant species in the 2/2 vineyards. While the vegetation was also dominated by annuals such as Lolium rigidum, Medicago minima, with a mean cover of 12.1 %, 7% respectively, perennials such as Lolium perenne (6.1 %) reached a much higher cover than in the other management types.

Predator communities

We captured 4 217 predators across the 26 vineyards. Ants were by far the most abundant predators (3 312 individuals), followed by spiders (631), ground beetles (152) and harvestmen (122). We observed an effect of inter-row management type only on ants abundance (Table 1, (Fig. 3). Only card position affected nocturnal predation (Table 2), and this effect was similar under the three different management types: larvae on the ground were more attacked than those on the vine trunk (0/2 : P < 0.001, 1/2 : P < 0.001 and 2/2 : P = 0.002; Table 2 and Fig. 3). Final predation significantly increased with increasing percentage of vegetation cover and species richness (PC1), both during the day and at night (Table 2).

The Cox-regression revealed a significant interaction between vegetation cover management and card position (Table 3, Fig. 4). Larvae on the ground were more susceptible to predation under all three management types. The predation risk for larvae both on the ground and on the vine trunk was 2.55 and 2.75 times higher in 2/2 than in 0/2 (respectively, P = 0.01 and P = 0.01). There was a significant effect of PC1 (percentage of vegetation cover and species richness), card position and their interaction on the survival time of sentinel prey. PC1 significantly affected the survival time of sentinel prey only on the vine trunk.

Observed predators

We observed and identified a predator in 80 % of the daytime predation events. Ants represented almost all the observations (96 %), with very limited predation by spiders (2.1 %) and other arthropods (1.9 %). Cataglyphis cursor was the most active ant on the ground, whereas Camponotus vagus and Crematogaster scutellaris were the most active predators on the vine trunk. Inter-row management type, card position and their interaction significantly affected the presence of ants on sentinel cards (Table 4). More ants were observed to attack larvae on the ground than on the vine trunk under 1/2 and 2/2 management types (Appendix C) (P < 0.001 for both). Significantly more larvae were attacked by ants in the 1/2 and 2/2 vineyards than in the tilled vineyards when on the ground (1/2: P = 0.02 and 2/2: P = 0.03). The model exploring the influence of vegetation parameters (PC1 and PC2), card

Discussion

This study reveals a clear positive effect of inter-row vegetation on the predation dynamics and final predation of sentinel prey. Although this effect was more pronounced on the ground, a spill-over from the vegetated inter-rows to the grapevines was observed. We also demonstrate for the first time in temperate perennial crops the predominant role of ants as potential natural enemies of pest insects. Our findings provide valuable insights into the strong functional role that vegetation plays as a driver of predation dynamics.

Predator response to vegetation cover

The predatory community was dominated by four taxa: ants, spiders, predatory ground beetles and harvestmen. These taxa recognised as generalist predators are commonly found in crop studies [START_REF] Symondson | Can generalist predators be effective biocontrol agents?[END_REF]. Ants were by far the most abundant predators, which is consistent with their ubiquitous occurrence and with other studies capturing ground-dwelling arthropods in Mediterranean vineyards [START_REF] Hoffmann | Can Flowering Greencover Crops Promote Biological Control in German Vineyards?[END_REF][START_REF] Carlos | How does habitat diversity affect ground-dwelling arthropods assemblages in Douro Demarcated Region terraced vineyards?[END_REF]Sáenz-Romo et al., 2019a). We find a management type effect only on ant abundance which were more abundant in fully vegetated vineyards than in tilled ones, and a response to vegetation parameters concerning PC2. Whereas ants did not significantly respond to the percentage of vegetation cover and species richness, they were less abundant in vineyards with high vegetation cover variance and diversity. Because ant abundance generally increases with an increase in vegetation structure complexity [START_REF] Ríos-Casanova | Ant diversity and its relationship with vegetation and soil factors in an alluvial fan of the Tehuacán Valley[END_REF] and is less affected by plant species composition and richness [START_REF] Uhey | Climate and vegetation structure shape ant communities along elevational gradients on the Colorado Plateau[END_REF][START_REF] Andersen | Responses of ant communities to disturbance: five principles for understanding the disturbance dynamics of a globally dominant faunal group[END_REF], but see Carlos et al., 2019), we would have expected a positive effect of an increase in variance of vegetation cover. It seems that vegetation structure can modulate ant responses to microclimate, which is a key driver in shaping ant communities [START_REF] Uhey | Climate and vegetation structure shape ant communities along elevational gradients on the Colorado Plateau[END_REF]. An increase in vegetation cover variance may have a global negative effect on the abundance of ants by decreasing soil humidity and increasing soil temperature. This is corroborated by the fact that fully vegetated vineyards, which has the lowest variance of vegetation cover, had almost twice the abundance of ants in 0/2 and 1/2 vineyards. Spiders and harvestmen were also affected by vegetation, their abundance increasing with increased vegetation cover variance and diversity only when vegetation cover and richness were high. While spiders are known to be affected by vegetation cover [START_REF] Costello | determine how habitat strata affects predator diversity and predation of Epiphyas postvittana (Lepidoptera: tortricidae) in a vineyard[END_REF][START_REF] Pfingstmann | Contrasting effects of tillage and Agriculture, Ecosystems and Environment 311[END_REF], this effect is less common for harvestmen [START_REF] Vogelweith | Cover crop differentially affects arthropods, but not diseases, occurring on grape leaves in vineyards[END_REF].

Effects of inter-row vegetation management on predation

Our results show that inter-row management type and percentage of vegetation cover, plant species richness (described by PC1) positively affected sentinel prey predation, in terms of both final predation and predation dynamics. The magnitude of this effect varied over compartments (ground vs grapevine plant) and periods (day vs night).

Overall diurnal attacks on sentinel prey both on the ground and on the grapevine plants increased from non-vegetated to fully vegetated vineyards, as a function of the increasing plant species richness and percentage of vegetation cover. At night, attacks on sentinel prey increased with plant species richness and percentage of vegetation cover but did not respond to inter-row management type. We observed an increase in predation events in both the 0/2 and 1/2 vineyards, which suggests a change in the predator community, with more predators being active at night. These communities may not respond in the same way to management. There is a scientific evidence that arthropods communities differ between day and night [START_REF] Petersen | Diel variation in the abundance and composition of the predator assemblages feeding on aphid-infested soybean[END_REF]. Importantly, the survival time was negatively correlated to vegetation cover indicating that predation events occurred faster in fully vegetated vineyards. A quick response to pest invasion is key to successful management of pest species [START_REF] Murdoch | Biological control in theory and practice[END_REF]. A positive relationship has previously been observed between local vegetation and predation occurring on the ground or on the crop [START_REF] Speight | The influence of weed-cover on the mortality imposed on artificial prey by predatory ground beetles in cereal fields[END_REF][START_REF] Danne | Effects of native grass cover crops on beneficial and pest invertebrates in Australian vineyards[END_REF][START_REF] Hoffmann | Can Flowering Greencover Crops Promote Biological Control in German Vineyards?[END_REF]. As suggested by [START_REF] Nicholls | Enhancing plant diversity for improved insect pest management in northern California organic vineyards[END_REF], non-crop vegetation increases the agroecosystem diversity and complexity providing habitats and resources for beneficial arthropods.

Predation occurring on the grapevine plants is important since predation on the ground does not necessarily provide protection against pest attack. Contrary to the results of Wilson et al. (2017a) not showing any effect on crop pests, we observed a spill-over effect from ground vegetation to vine trunk. Sentinel preys on the trunk were attacked more often in fully vegetated vineyards than in vineyards with bare soil. [START_REF] Irvin | The effect of an irrigated cover crop on grape vine productivity, and beneficial insect and grape pest abundance in southern California[END_REF] showed that migration of pests and beneficial insects into the grapevine canopy was enhanced by vegetation cover. They suggested that buckwheat is highly attractive to beneficial insects, with a secondary migration from ground vegetation to the crop. Movements from ground vegetation to the canopy of a pear orchard were studied using an immunomarking method [START_REF] Horton | Use of a new immunomarking method to assess movement by generalist predators between a cover crop and tree canopy in a pear orchard[END_REF] to determine the possible contribution of vegetation cover to the crop canopy predator community. Their findings confirmed observations by [START_REF] Frank | Video analysis to[END_REF] and [START_REF] Shapira | Habitat use by crop pests and natural enemies in a Mediterranean vineyard agroecosystem[END_REF] showing that the community of generalist predators in the crop canopy differs from that of predators in the ground vegetation. These findings may explain the differences between final predation and the survival times of sentinel prey on the ground and on the vine trunk observed in our study. Natural enemies have been found to be more abundant and more active in the ground vegetation than in vine foliage because of its higher diversity of habitats and resources [START_REF] Frank | Video analysis to[END_REF][START_REF] Shapira | Herbaceous vegetation enhancement increases biodiversity in a wine-producing vineyard in Israel, promoting shifts in agricultural practices in other vineyards[END_REF].

The unexpected role of ants in predation

A positive correlation between non-crop vegetation cover and predator densities and a corresponding negative correlation with pest insect densities have been demonstrated in other studies [START_REF] Nicholls | Reducing the abundance of leafhoppers and thrips in a northern California organic vineyard through maintenance of full season floral diversity with summer cover crops[END_REF][START_REF] Vogelweith | Cover crop differentially affects arthropods, but not diseases, occurring on grape leaves in vineyards[END_REF]. In our case, the responses of predation and of ant abundance in traps to inter-row management were correlated.

In the almost 80 % of predation events for which we identified a predator preying on larvae, 96 % were ant workers. We observed two different ant communities attacking sentinel prey, one in the inter-rows and the other in the grapevine canopy. Predation events on the ground were dominated by Cataglyphis cursor, whereas Crematogaster scutellaris and Camponotus vagus were the predominant species on the vine trunk. Ants are among the most abundant surface-active arthropods captured in agrosystems [START_REF] Carlos | How does habitat diversity affect ground-dwelling arthropods assemblages in Douro Demarcated Region terraced vineyards?[END_REF]Sáenz-Romo et al., 2019a). They have previously been found to attack sentinel prey in orchards and agroforestry systems [START_REF] Mathews | Habitat manipulation of the apple orchard floor to increase ground-dwelling predators and predation of Cydia pomonella (L.)(Lepidoptera: tortricidae)[END_REF][START_REF] Martin-Chave | Agroforestry has an impact on nocturnal predation by ground beetles and Opiliones in a temperate organic alley cropping system[END_REF], but never in vineyards [START_REF] Danne | Effects of native grass cover crops on beneficial and pest invertebrates in Australian vineyards[END_REF]. Vineyard inter-rows are more exposed to high temperatures and drought events than orchard inter-rows. In our study, the high abundance of C. cursor, a hot climate specialist, is consistent with theses microclimatic differences between vineyards and orchards. The relatively warm and dry conditions of our vineyards may also explain the low abundance of predatory ground beetles commonly found in orchards [START_REF] Mathews | Habitat manipulation of the apple orchard floor to increase ground-dwelling predators and predation of Cydia pomonella (L.)(Lepidoptera: tortricidae)[END_REF].

Overall predation by ants was affected by inter-row management type and card position. When cards were on the ground, more ants were observed in both types of vegetated vineyards than in non-vegetated vineyards. The surface-active C. cursor was the only species that position and their interaction on observations of ants attacking larvae revealed a significant effect of card position only (Table 4).

When the three species were modelled separately, Cataglyphis cursor was the only species affected by inter-row management type, attacking larvae more in the vegetated inter-row vineyards (1/2: P = 0.01 and 2/2: P < 0.05). The number of Camponotus vagus observations was dependent on the interaction between PC1 and PC2 (Table 4). choice, establishment and management of appropriate inter-row vegetation [START_REF] Celette | Competition for nitrogen in an unfertilized intercropping system: the case of an association of grapevine and grass cover in a Mediterranean climate[END_REF][START_REF] Ruiz-Colmenero | Soil and water conservation dilemmas associated with the use of green cover in steep vineyards[END_REF]. Choosing plant species that improve desired ecosystem services and reduce disservices based on plant functional traits is one possible way to address such constraints [START_REF] Gardarin | Understanding plant-arthropod interactions in multitrophic communities to improve conservation biological control: useful traits and metrics[END_REF]. To our knowledge, our study is the first to demonstrate the key role of ants in insect predation and the influence of inter-row vegetation on predatory ants in Mediterranean vineyards. Effects on ants should thus be considered in plant species selection and vegetation management. responded to inter-row management type, with more attacks in the two vegetated management types than in the tilled one. Ants are known to be sensitive to tillage [START_REF] Sharley | The effects of soil tillage on beneficial invertebrates within the vineyard[END_REF], which, together with an increase in available resources, may explain why ground predation increased in vegetated inter-rows whereas vine trunk predation by ants was unaffected by management type. However, more observations of one of the predominant predators on the vine trunk, i.e. C. vagus, were made on vine trunks in fully vegetated vineyards (in total, 50 observations of predation events) than in tilled vineyards (in total, 27 observations of predation events). They were influenced by the interaction between PC1 and PC2 (species richness and percentage of vegetation cover, and percentage of vegetation cover variance and community diversity, respectively) suggesting an influence of inter-row vegetation on vine trunk ants.

Although they are considered as generalist predators [START_REF] Symondson | Can generalist predators be effective biocontrol agents?[END_REF], the role of ants in biological control under temperate and Mediterranean conditions is often underestimated due to their putative negative effect on the crop through mutualistic interactions with aphids and coccids [START_REF] Beltrà | Association between ants (Hymenoptera: formicidae) and the vine mealybug (Hemiptera: pseudococcidae) in table-grape vineyards in Eastern Spain[END_REF]. In tropical regions, ants are widely recognised as natural biocontrol agents, known to regulate pest populations [START_REF] Philpott | Biodiversity in tropical agroforests and the ecological role of ants and ant diversity in predatory function[END_REF][START_REF] Offenberg | Ants as tools in sustainable agriculture[END_REF]. It has been reported that Solenopsis germinata preys on eggs and larvae of the Asian corn borer, Ostrinia furnacalis, which might reduce pest infestation [START_REF] Litsinger | Maize planting time and arthropod abundance in southern Mindanao, Philippines. II. Population dynamics of natural enemies[END_REF]. Similarly, [START_REF] Sunamura | Predation of invasive rednecked longhorn beetle Aromia bungii (Coleoptera: cerambycidae) eggs and hatchlings by native ants in Japan[END_REF] found that native Japanese ant fauna may increase resistance to Aromia bungii attack in Rosaceae trees. The potential of biological pest control by ants requires further investigation and should be considered in the management of vineyards in regions where aphids and coccids do usually not affect vineyards.

Implications for vineyard management

Winegrowers use the vegetation cover to increase the load-bearing capacity of the soil, allowing machinery use regardless of the rain. Vegetation cover also reduces erosion and runoff. This practice is becoming increasingly common, in line with biodiversity-friendly practices such as organic and biodynamic farming. This nature-based solution is the subject of an increasing number of scientific studies that have highlighted its importance in restoring or improving the multifunctionality of vineyard ecosystems and the ecosystem services they provide [START_REF] Winter | Effects of vegetation management intensity on biodiversity and ecosystem services in vineyards: a meta-analysis[END_REF]. Our study provides evidence that vegetation cover enhances predation by generalist arthropods in Mediterranean vineyards.

Enhancing the predation function improves vineyard resilience to pest invasion, protecting the crop at the pest arrival stage and decreasing establishment success. Although we did not specifically evaluate the pest regulation service, generalist predators may rapidly and efficiently act as a shield to protect the vineyards from pests such as vine budworms. Because of their high abundance and diversity of foraging strategies, the presence of ants increases the probability of pest detection, which is crucial in pest regulation. The two main predators identified on vineyard plant, C. vagus and C. scutellaris are active from April to October, which corresponds to that of the two pest insects L. botrana and S. titanus. They live in colonies of several hundreds to thousands of individuals, and have complementary foraging habits. Camponotus vagus, one of the largest ants in Europe (worker size ranges from 6 to 14 mm), which has a solitary to sometimes group foraging strategy is able to attack prey larger than 2 mm. Crematogaster scutelaris has smaller workers (approximately 6 mm) and compensate for a size deficit by the number of individuals attacking a prey.

Restoring vegetation cover will also lead to a reduction in herbicide use and/or tillage and may reduce insecticide treatments. However, several constraints still prevent full adoption of vegetated inter-rows, such as competition for water and other resources. In the Mediterranean region, the water stress common during summer limits grapevine growth, yield and grape quality, and thus represents a major factor in the

Fig. 1 .

 1 Fig. 1. Position of analysed wineries in the southern Luberon area (south-eastern France). In each winery, one to four vineyards were involved in the study.

Fig. 2 .

 2 Fig. 2. Example of position of pitfall traps (white circles) and vegetation quadrats (white squares) in a partially vegetated vineyard (1/2).

Fig. 3 .

 3 Fig. 3. Mean final predation rates (± SE) on sentinel prey per inter-row management type during day A. and at night B. Different letters indicate significant differences between management types (P < 0.05), capital letters are used to compare observations on the vine trunk and lowercase to compare observations on the ground. Stars are used to compare observations between the two compartments within management types (ns: non-significant; **:P<0.001; ***: P < 0.0001).

Fig. 4 .

 4 Fig. 4. Effect of management type on survival of larvae (sentinel prey) on the ground and on vine trunk during daytime (Kaplan-Meier method). Different letters indicate significant differences in pairwise comparisons using the Coxmodel (P < 0.05). Capital letters are used to compare observations on the vine trunk and lowercase to compare observations on the ground. Stars are used to compare observations between the two compartments within management types (dotted line: tilled vineyards (0/2); dashed line: partially vegetated vineyards (1/2); solid line: fully vegetated vineyards(2/2); (*: P < 0.01; nonsignificant; ***: P < 0.0001).

Table 1

 1 Effects of inter-row management and vegetation parameters on recorded predator taxa in pitfall traps. PC1 is the PCA first axis related to percentage of vegetation cover and plant species richness. PC2 is the second axis related to beta diversity and vegetation cover variance. The different models are separated by line. Only significant interactions (<0.05) are indicated in the table. Values in bold indicate significant effects at P < 0.05.

	Response variables	Explanatory variable	Estimate / Chisq	P-value
		Management type	7.22	0.03
	Ant abundance	PC1	3.79	0.05
		PC2	3.95	<0.05
		Management type	0.15	0.93
	Spider abundance	PC1 PC2	2.94 0.00	0.09 0.98
		PC1*PC2	7.69	0.01
		Management type	2.03	0.36
	Carabidae abundance	PC1 PC2	0.14 0.05	0.71 0.83
		PC1*PC2	3.81	0.05
		Management type	0.84	0.66
	Harvestman abundance	PC1 PC2	0.02 0.01	0.90 0.93
		PC1*PC2	7.10	0.01

Table 2

 2 Summary of the generalized linear mixed-effects models explaining final predation during day and at night. PC1 is the PCA first axis related to percentage of vegetation cover and plant species richness. PC2 is the second axis related to beta diversity and vegetation cover variance. The different models are separated by line. Only significant interactions (<0.05) are indicated in the table. Values in bold indicate significant effects at P < 0.05.

	Response variables	Explanatory variables	Chisq	P-value
		Management type	9.39	0.01
	Diurnal predation	Card position	62.74	<0.001
		Management type*Card position	9.97	0.01
	Nocturnal predation	Management type Card position	3.65 68.90	0.16 <0.001
		PC1	4.95	0.03
	Diurnal predation	PC2	0.23	0.63
		Card position	62.80	<0.001
		PC1	5.18	0.02
	Nocturnal predation	PC2	0.01	0.92
		Card position	68.94	<0.001

Table 3

 3 Effect of inter-row management type and card position and their interaction on survival time of sentinel prey according to the Cox-model. PC1 is the PCA first axis related to percentage of vegetation cover and plant species richness. PC2 was removed following model selection. The different models are separated by line. Only significant interactions (<0.05) are indicated in the table. Values in bold indicate significant effects at P < 0.05.

	Response variable	Explanatory variables	Chisq	P-value
		Management type	8.07	0.02
		Card position	40.96	<0.001
	Survival time of sentinel prey	Management type*Card position PC1	7.54 4.01	0.02 <0.05
		Card position	40.54	<0.001
		PC1*Card position	8.94	0.003

Table 4

 4 Summary of the generalized linear mixed-effects models explaining observations of ants attacking sentinel prey. Only cards on the ground were considered for C. cursor and only cards on the vine trunk were considered for both C. vagus and C. scutellaris. PC1 is the PCA first axis related to percentage of vegetation cover and plant species richness. PC2 is the second axis related to beta diversity and vegetation cover variance. Only significant interactions (<0.05) are indicated in the table. Values in bold indicate significant effects at P < 0.05.

	Response variables	Explanatory variables	Chisq	P-value
		Management type	4.48	0.11
		Card position	61.83	<0.001
	Observed ant attacking prey	Management type*Card position	9.28	0.01
		PC1	3.27	0.07
		PC2	0.02	0.89
		Card position	61.83	<0.001
	Observed C. cursor attacking prey	Management type PC1 PC2	8.90 1.81 0.01	0.01 0.18 0.93
		Management type	2.50	0.29
	Observed C. vagus attacking prey	PC1 PC2	1.18 0.01	0.28 0.93
		PC1*PC2	5.84	0.02
	Observed C. scutellaris attacking prey	Management type PC1 PC2	4.03 0.30 0.38	0.13 0.58 0.54

Acknowledgements

We want to express our gratitude to the cooperative of Marennon and all the farmers for authorising us to access to their vineyards and for their interest in the project. Thanks to Daniel Pavon, Emile Melloul, Elza Boniffacy, Pauline Mathieu and Romain Septidi-Lenfant for valuable help during the fieldwork.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Declaration of Competing Interest

The authors report no declarations of interest.

Appendix A. Supplementary data

Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j.agee.2021.107327.