Yun Lu
email: yun.lu1@tu-dresden.de

Christoph Statz

Sebastian Hegler

Wolf-Stefan Benedix

Valérie Ciarletti
email: valerie.ciarletti@latmos.ipsl.fr

Dirk Plettemeier

Multiple Compressive Projection Measurement for Stepped Frequency Radar

, which is a subtle mathmatic application in practice and leads to a great sucess in signal detection both for communication and measurement, e.g. radar technique. In spite of this radical progress there are still a lot of open problems. One of them is the "noise" including background noise and nonideal signal modelling, which is not just a problem for CS but a general difficulty for signal processing. Although there are many sophisticated recovery algorithms developed to cope with noise, the performance will be usually impacted by inaccurate noise estimation or modelling error. In this paper, we will analytically describe the multiple compressive projection measurement (mCPM 1 or MCPM) introduced in [2]. Both theoretical analysis and numerical evaluations show that mCPM is a promising measurement system.

I. INTRODUCTION

Suppose we have some target signal spanned in a particular frame Φ and provide a generative form

t =Φw + ξ, (1)
where t ∈ R n is the vector of targets, Φ ∈ R n×d is a frame of d diversities that have been assumed, w is a vector of unknown weights, and ξ is Gaussian noise. The goal is to estimate w given t and Φ. Obviously, the maximum likelihood (ML) solution is very effective, if the dimension of w is smaller than that of the signal, i.e. n>d . In case of n<d , which is said to be underdetermined, a sparse w is required for a unique solution. Those problems can be handled by a general Bayesian framework [START_REF] Ji | Bayesian compressive sensing[END_REF]. An accurate result can be obtained if precise Φ and a reasonable estimation about ξ are available. Unfortunately, this is usually not the case in practice. That means, mostly we get only an estimation about Φ, denoted as Φ, as well as ξ. Thus, even without noise ξ we still cannot get the exact solution. To combat the noise ξ, which is usually nonsparse in Φ, we introduced the mCPM. By using the restricted isometry property (RIP) for both sparse and non-sparse signal as well as the multi-correlation function (MCF) [START_REF] Lu | Novel multi-correlation differential detection for improving detection performance in DSSS[END_REF], the noise term in (1) can be well suppressed without exact information about the noise level in the channel. This work will focus on 1 The special case is m =1as the conventional CS.

the general signal sensing2 of stepped frequency radar (SFR) detecting sub-surface objects.

A. Organisation

The remaining of the paper is organised as follows: The Section II introduces the property of underdetermined linear systems und their solutions. Then, the theoretic detection behavior in compressive domain will be presented in Section III. In Section IV we will introduce the MCF as well as its combination with CS, termed as mCPM. The basic properties of mCPM will be discussed. After that, we apply the mCPM in SFR and formulate new measurement approach for better performance. Finally, the summary and future work are presented.

II. UNIQUE SOLUTION

Observing (1) with n<d , there are infinite solutions due to the fact that there are some linearly dependent diversity columns in Φ. In [START_REF] Donoho | Optimally Sparse Representation from Overcomplete Dictionaries via l1-norm minimization[END_REF] the author gave a sparse condition in the sense of linear algebra for a unique solution, which was derived by clearly distinguishing two different unknown weights w 1 and w 2 . That means, the difference of δ = w 1 -w 2 must be uniquely detectable with respect to [START_REF] Candes | Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information[END_REF]. In other words, δ is not allowed to lie in the null space of Φ, i.e. Φδ =0. Thus, the columns controlled by δ, i.e. Φ δ ∈ R n×m δ and m δ ≤ n, have to be linearly independent. Because of the randomness of δ in practice, Φ δ ranges over the whole Φ. Finally, the number of non-zero entries in δ, denoted as δ 0 := |supp(δ)| and supp(δ)={j : δ j =0 }, have to be smaller than the spark, which denotes the least number of linearly dependent columns in Φ. Formally,

spark(Φ) = min w =0 w 0 subject to Φw =0. (2)
Furthermore, observing δ, we will get the maximal δ 0 if weights w 1 and w 2 are disjoint (extreme case). This means

δ 0 ≤ w 1 -w 2 0 = w 1 0 + w 2 0 for w 1 ∩ w 2 = ∅. (3)
Obviously, (3) must hold the following condition

max{ δ 0 } <spark(Φ). (4)
Thus, for a general unknown weight w and w 0 = K, we have max{ δ 0 }≤2K<s p a r k (Φ).

(5)

At last, the unique solution is given by holding the condtion K<s p a r k (Φ)/2 and w 0 = K.

The results in [START_REF] Johnson | Extensions of lipschitz mappings into a hilbert space[END_REF] can also be interpreted as: The vector δ still preserves its Euclidean distance approximately after linear projection, i.e. Φδ. In other words, it holds

Φδ 2 2 ≈ Iδ 2 2 , ( 7
)
where I is the identity matrix. This means, the 2K-th normalized singular value of Φ is not far from unit, which essentially requires that Φ δ behaves like an orthogonal system. Similar results can also be found in Johnson-Lindenstrauss (JL) lemma [START_REF] Johnson | Extensions of lipschitz mappings into a hilbert space[END_REF]. The JL lemma is concerned with the following problem. Given a set of points in R d , we would like to embed these points into a lower-dimensional Euclidean space R n (n<d ) while approximately preserving the relative distances between any two of these points. Let ∈ (0, 1) be given. For every set

Q of #(Q) points in R d , if n is a positive integer such that n>n 0 = O(ln(#(Q))/ 2 ), there exists a Lipschitz mapping f : R d → R n such that (1 -) ≤ f (u) -f (v) 2 l n 2 u -v 2 l d 2 ≤ (1 + ) (8)
for all u, v ∈ Q. That means, the Lipschitz function f as an injective function and the corresponding l 2 -norm distance ratio is bounded by 1+ . The JL lemma leads directly to the restricted isometry property (RIP) in CS [START_REF] Candes | Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information[END_REF]: If matrix Φ ∈ R n×d satisfies the RIP of order K with a constant ∈ (0, 1), such that (1 -) w

2 2 ≤ Φw 2 2 ≤ (1 + ) w 2
2 holds for all w ∈ K , then w is recoverable. Usually, w can be obtained by maximum a posteriori (MAP) estimation like l pnorm regularization and FOCUSS [START_REF] Barrodale | Applications of mathematical programming to lp approximation[END_REF] solving:

min Φw -t 2 2 + λ w p p for 0 ≤ p ≤ 1. (9)
In addition to MAP estimation, the full Bayesian approach [START_REF] Ji | Bayesian compressive sensing[END_REF], e.g. sparse Bayesian learning (SBL), which seeks for the distribution mass, improves the detection performance dramatically. The basic cost function can be given as:

L = log |Σ t | + t T Σ -1 t t
, where Σ t = σ 2 I + ΦΓΦ T with hyperparameters Γ=diag(γ) controlling the variance of entries in w. However, its performance is still very sensitve to inaccurate noise estimation. In the next section we will introduce the mCPM method to cope with "noise" effect.

III. COMPRESSIVE DETECTOR AND mCPM

A. Compressive Detector

To improve the detection performance in underdetermined framework, we have to exame the statistic behavior in compressive domain with respect to the original signal domain.

For example, to distinguish two hypotheses:

H 0 : y = Mξ H 1 : y = M (x + ξ) (10)
where M ∈ R mM ×n with m M <nis a measurement matrix, ξ ∼N(0,σ 2 I d ) is i.i.d. Gaussian noise. Then, the distribution of y can be given as

f 0 (y)= exp -1 2 y T Σ M y |σ 2 MM T | -1/2 (2π) mM /2 f 1 (y)= exp -1 2 (y -Mx) T Σ M (y -Mx) |σ 2 MM T | -1/2 (2π) mM /2 (11)
where

Σ M =( σ 2 MM T ) -1 .
Based on the likelihood ratio test Λ(y)= f1(y) f0(y) ≷ H1 H0 η, where η is given for particular false alarm probability α = Λ(y)>η f 0 (y)dy, we can get an equivalent detector in a compressive domain 3 :

t := y T (MM T ) -1 Mx. ( 12
)
It is easy to show that

tH0 ∼N 0,σ 2 x T P † M Mx tH1 ∼N x T P † M Mx,σ 2 x T P † M Mx . (13)
where If x =Φ w and Φ=I d×d , the detection performance in compressive domain is strongly depending on M . For a compressive detector, i.e. M is a wide matrix, x must live in the row span of M with high probability, which is equivalent to require a sparse x. Furthermore, for a Gaussian matrix

P † M = M T (MM T ) -1 . -0.5 0 0.5 1
M i ∈ R mM ×n , x T P † Mi M i x is highly concentrated around mM n x 2 2 . That is, E x T P † Mi M i x = m M n x 2 2 . (14)
Finally, with high probability we get the detection performance [9]

P D (α) ≈ Q Q -1 (α) - ŜNR (15) with ŜNR = m M n SNR and SNR = x 2 2 /σ 2 , and Q(z)= ∞ z exp(-u 2 /2)du. 1)
Influence of compressive projection: Observing (15) the detection performance will be strongly affected by the dimension ratio of M , i.e. m M /n. In case of m M /n =1 , the compressive projection is just an implementation of a traditional dectector (matched filter (MF), signal based). Then, we obtain the well-known detection performance:

P D (α) ≈ Q Q -1 (α) - √ SNR . ( 16
)
For the case of m M /n < 1, we could still get good recovery performance, if x lies in row span of M with high probability (vector based, e.g. CS, deteriorated detection performance depending on m M /n but still detectable).

2) Advantage of compressive projection: Compressive projection is not alwalys negative for information detection. As we know, the basic feasible solution to ( 9) is sparse, i.e. w 0 ≤ n. However, in some cases like strong noise and coherent signals, the resulting solution still cannot avoid radical over-fitting. Although it can be partly compensated by choosing proper λ, the performance remains too sensitive to inaccurate λ. Besides, one is not allowed to set the λ too large (otherwise w disappears). Alternatively, observe the sparsity bound of solution from [START_REF] Scharf | Statistical Signal Processing: Detection, Estimation, and Time Series Analysis[END_REF], it is limited by the row dimension n of Φ. In other words, we can reduce the row dimension to be an appropriate value in need for avoiding radical over-fitting regardless of recovery algorithms. Mathematically,

min M (Φw -t) 2 2 + λ w p p for 0 ≤ p ≤ 1. (17)
If x lies in the row span of M in high probability, the obtained sparser solution (local optimal solution) usually includes weights w or at least part of them (in critical situation). This sparsity controlling, i.e. w 0 ≤ m M <nwith M ∈ R mM ×n and Φ ∈ R n×d , excludes the over-fitting in unfavorable situation dramatically, since over-fitting usually needs particular amount of non-zero entries in solution. Thus, we expect the compressive detector having better performance against over-fitting than that of traditional detector in particular circumstance, namely requiring that reduction of n will not result in severe detection performance deterioration 4 . Yet, a solution without strong over-fitting is not enough for stable detection. Recall that the sparser solution by compressive detector has possibly only part of weights w. To cope with this problem we will introduce the principle of multi-correlation function (MCF). 4 The row dimension of matrix gives an upper bound of sparsity of feasible solution regardless of recovery algorithms. To control sparsity one can also limit the number of iterations of sparse recovery algorithm (however, it sometimes hampers the convergence). An alternative way for over-fitting controlling is noise mitigation by column dimension extension. In this paper we mainly consider the case of compressive projection, i.e. row dimension reduction.

B. MCF

The MCF is basically an extension of the MF for achieving better correlation properties. The prime idea of MCF is based on the delay and multiply (DAM) property of m-code, which was later extended for generation of Gold codes. The DAM of m-code as well as Gold codes indicates that a transform of one m-code to its other phase delay or a transform of one Gold code to other family members can be realized by applying the DAM operation. Later, this property was applied to other Galois field (GF)-based codes. More information can be found in [START_REF] Lu | Method and apparatus for the differential multiple correlation in broadband ultrasonic location systems[END_REF]. A paralell combination of several different correlation functions can result in much better correlation property than 1/ √ N given by the Sarwate lower bound, where N is the sequence length. Thus, it is very favorable for signal detection. Formally, the MCF can be given as

C(s)= 1 m m k=1 F k = 1 mN m k=1 N -1 i=0 I k (u i )I * k (v i+s ), ( 18
)
where m is the number of combinations, u and v are the original input sequences and the reference sequences, respectively. The code transform is given by the transform function I, e.g. DAM operation as above. The term F k in (18) could be any kind of "correlation" process or more generally recovery algorithm. The objective of combination of m paralell results from F k is achieving coherent combination or collecting the partial result from single F k with respect to information and non-coherent combination of "noise" such that an even better detection scenario.

C. mCPM

The introduction of MCF into compressive detector, which is termed as mCPM here, could be very promising in our particular case and can also facilitate parameter adjustment with respect to information recovery in practice. Basically, mCPM consists of two steps: i) the first phase is actually a normal compressive projection and recovery (CPR) process by algorithm promoting sparse solution just as (17); ii) and the second phase is iteratively updating (combining) sparse solutions from each CPR by different compressive projections, e.g. different measurement matrix M i , i.e. Θ i = M i Φ, i denotes the i-th iteration. Due to the underdetermined property y = Mx =Θ w with Θ ∈ R n×d and n<das well as Θ holding the RIP with respect to w in high probability, the sparse solution ŵ by single CPR includes w with high probability while it provides randomness of noise in terms of their amplitudes and positions, since noise is usually nonsparse. Therefore, a combination of a series of ŵi

w g = m i=1 ŵi (19)
can result in coherent combination of information components w and non-coherent combination of noise and thus a favorable detection scenario. Finally, the two close distribution modes of CS detector in Fig. 1 Obviously, the performance of mCPM is strongly depending on the compressive projections. An effective way for noncoherent combination of noise term in (19) requires that all compressive projection matrice M i should be less correlated with each other. Otherwise, the recovered noise term from two different iterations are similar and will also be coherently combined by (19), i.e. yield no contribution for distance expansion between signal term and noise term. Fig. 4 presents the estimation error e = w gw 2 / w 2 by mCPM with m = 20 depending on row dimension of M . In case of small row dimension, the estimation error is relatively large, since the resulting reference matrix Θ does not hold the RIP condition with respect to w. For relatively large row dimension of M it also provides increased estimation error. The reason, as aforementioned, is that except the signal components a lot of recovered noise components also exhibit independence of random M . This means, there are many noise components also live in the row span of the resulting reference matrix Θ, i.e. inefficiency of non-coherent noise combination. Besides, we

Θ i ∈ R 30×200 , m =3 0and for MF Θ ∈ R 200×200 , sparsity w 0 =2, SNR=10dB.
can freely determine the number of projections m in mCPM for particular performance (see. Fig. 5). Setting a relatively large estimated noise factor the error by MF decreases fast, however, is not recommended, which is equivalent to set large λ in ( 9) and possibly results in the loss of information. This problem can be solved very well by mCPM, since mCPM still provides relatively small estimation error at low estimated noise factor (see Fig. 6).

IV. mCPM FOR STEPPED FREQUENCY RADAR (SFR)

A. SFR using CS

In the SFR radar [11], it observes the scene with a discrete set of frequencies and synthesizes the impulse in the frequency domain, and brings advantage of better accuracy. They pointed out that CS can be done in frequency domain by randomly measuring all Fourier coefficients. Thus, for rough detection it requires only a small amount of frequency measurements, which can first reduce the measurement time, and second save energy such that long-term activity is possible.

In this paper, we collect the frequency measurements from real circumstance and devices and process them directly in frequency domain. The connection to time domain for ranging information is simply the IFFT transform. Our objective vector, which indicates the ranging information, is w t . Its corresponding vector in frequency domain is w f = Fw t , where F ∈ C d×d is the FFT matrix and the dimension d determines the required resolution in time domain. The practical measurement for SFR is basically conducted in frequency domain and the number of frequency measurement points n is usually less than d. Thus, the actual obtained frequency By using CS, we can futhermore reduce the number of frequency measurements per projection matrix M . Therefore, an underdetermined linear system can be constructed as

ŵfr = M • R • F • w t =Θw t , (20)
where M determines which frequency point should be active (see Fig. 7). Finally, one needs to seek w t by solving Relative error normed on the recovery error at sampling 10 frequencies.

convex optimization can be realized by FOCUSS or SBL approach. By given prior knowledge that the most reflections are clustered together we can limit the column dimension of Θ for better performance. Fig. 8 presents the relative recovery error by different number of frequency measurements. We can notice that it requires only about 50 frequencies for rough detection. In case of frequency number less than 30 the error increases dramatically. In practice, the SFR can work with following principle: For rough detection the radar can randomly (if necessary) and quickly measure just a few number of frequency points within a defined frequency range as presented in Fig. 7; In case of fine detection, the radar usually tries to collect as many frequency measurements as possible.

B. Full frequency measurement by mCPM

However, how to collect these frequency measurements plays a major role for stable and accurate information recovery. According to the conventional collection scheme, all defined frequency points will be measured once and processed. This approach, however, suffers from non-ideal signal modelling and inaccurate noise estimation especially for recovery algorithms, which are very sensitive to those effects (see Fig. 9).

Alternatively, the radar can work in mCPM mode, i.e. frequency measurements are collected partly, randomly and iteratively. Results from each iteration will be combined. This work mode, as discussed above, provides very stable recovery performance in a non-ideal scenario and facilitates parameter adjustment (see Fig. 10). The results in Fig. 9 and 10 are in the case that the influence of antenna pattern has not been calibrated. Nevertheless, the performance by mCPM is still well and better than the conventional work mode.

V. SUMMARY

In this paper we investigate the stepped frequency radar signal acquisition and processing by CS (or by generalized mCPM). SFR using CS is a very promsing approach for longterm activity. By using mCPM5 , which can facilitate the fine measurement in practice, it provides even better performance than the conventional work mode. Basically, this work is not considering the feasibility and advantages of CS, rather a stable detection and estimation in noisy situation. The above results shows that even in full frequency mode, the mCPM mode is recommended.

As future work, we will focus on better energy distribution in frequency domain (frequency coding) such that giving better correlation properties in particular area in time domain (

) 21
where Ψ=[ Φ ,R] and M6 , R could be random matrix.

The final solution is given by pruning the w, i.e. w ∝ wΦ , where wΦ is the subvector related with Φ. The NM method is expected to be more stable than row dimension reduction, since the change of row dimension is directly proportional to the recoverable sparsity, while the variation of column dimension is effected by a logarithm factor. More details will be discussed in our forthcoming paper.

[

Figure 1 .

 1 Figure 1. Distribution of hypotheses in detection domain by CS and MF.

Figure 3 .

 3 Figure 3. SNR=3dB, w 0 =2 , MF with Θ ∈ R 200×200 , mCPM with Θ i ∈ R 50×200 and m =10 the false alarm probability (Pf) and detection probability (Pd) depending on different detection thresholds by normalizing the values within [0, 1]. Indeed, lines of Pf and Pd by mCPM are more favorable than that by MF.Obviously, the performance of mCPM is strongly depending on the compressive projections. An effective way for noncoherent combination of noise term in (19) requires that all compressive projection matrice M i should be less correlated with each other. Otherwise, the recovered noise term from two different iterations are similar and will also be coherently combined by (19), i.e. yield no contribution for distance expansion between signal term and noise term. Fig.4presents the estimation error e = w gw 2 / w 2 by mCPM with m = 20 depending on row dimension of M . In case of small row dimension, the estimation error is relatively large, since the resulting reference matrix Θ does not hold the RIP condition with respect to w. For relatively large row dimension of M it also provides increased estimation error. The reason,

Figure 4 .Figure 5 .

 45 Figure 4. Sparsity w 0 =4, SNR=10dB, Θ=M Φ and Φ ∈ R 200×200

Figure 6 .

 6 Figure 6. Reference matrix for mCPMΘ i ∈ R 30×200 , m =3 0and for MF Θ ∈ R 200×200 , sparsity w 0 =2, SNR=10dB.

Figure 7 .

 7 Figure 7.One practical example for randomly sub-sampling only 67 measurements in frequency domain, there are totally 1001 frequencies range from 500MHz to 3GHz.vector isw n×1 fr = Rw d×1 f , where R =[ I n×n , 0 n×(d-n) ].By using CS, we can futhermore reduce the number of frequency measurements per projection matrix M . Therefore, an underdetermined linear system can be constructed as

min{λ w t p p + ŵfr -Θw t 2 }

 2 with 0 ≤ p ≤ 1. This non-

Figure 8 .

 8 Figure 8.Relative error normed on the recovery error at sampling 10 frequencies.

2 2

 2 . The results above are just based on the random selection principle. An optimal selection is still an open problem and also problem dependent. Further research direction is the noise mitigation (NM) by reference matrix extension. Generally [12], min M (Ψ wt) + λ w p p for 0 ≤ p ≤ 1.

Figure 9 .

 9 Figure 9. Recovery by full-sampling (1001 frequency points), MF+SBL, estimated noise factor is 0.5 × 10 -1 . Due to non-ideal noise estimation the performance is moderate

Figure 10 .

 10 Figure 10. Recovery by randomly sub-sampling (average frequency points for each iteration is 80), mCPM+SBL with m =3 0 , estimated noise factor is 0.5 × 10 -1 . Recovery condition was same as in Fig.9.

  would be pulled apart from each other as well as their variance can be decreased depending on M and m. A schematic illustration of mCPM is presented in Fig.2. As a result, the radical over-fitting of the solution

				received datas;	reference matrix
					compressive projection
					sparse signal recovery
						solution update
						stop condition ?		no
							yes
							end
						Figure 2. mCPM
	can be controlled by compressive recovery while its estimation
	deterioration is compensated by MCF principle. Fig. 3 shows
		1
		0.9
		0.8
		0.7					Pd, MF+SBL
	probability	0.4 0.5 0.6					Pf, MF+SBL Pd, mCPM+SBL Pf, mCPM+SBL
		0.3
		0.2
		0.1
		0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
							threshold

  11] A. B. Suksmono; E. Bharata; A. A. Lestari "A compressive SFCW-GPR system". in proc. of 12th Int Conf. on Ground Penet Radar, Birmingham, pp. 16-19, 2008. [12] Y. Lu, C. Statz, M. Mütze, S. Hegler, D. Plettemeier "Noise Mitigation and Extraction of Scatters in Acoustic Imaging by Compressed Sensing", XVIII-th International Seminar/Workshop direct and inverse problems of electromagnetic and acoustic wave theory, Lviv, Ukraine, 2013.

It is not necessarily in sub-Nyquist framework.

y T (MM T ) -1 Mx ≷ H 1 H 0 σ 2 log(η)+ 1 2 x T M T (MM T ) -1 Mx. In case of M = I, then y T x ≷ H 1 H 0 σ 2 log(η)+ 1 2 x T x.

It mainly focuses on stability by modifying the cost function.

M can be identity matrix or random matrix.

Acknowledgment: WISDOM development and this research were supported by funding from the French space agency CNES and from the German space agency DLR.


Hui Zhang
email: hui.zhang@mailbox.tu-dresden.de

Wolf-Stefan Benedix
email: wolf-stefan.benedix@tu-dresden.de

Dirk Plettemeier
email: dirk.plettemeier@tu-dresden.de

Valérie Ciarletti

Radar Subsurface Imaging by Phase Shift Migration Algorithm

In this paper the phase shift migration based Synthetic Aperture Radar (SAR) is described and applied on radar imaging for dual polarized ground penetrating radar system (GPR). Conventional techniques for SAR imaging focusing use the matched filter concept and convolve the measurement data with a filter impulse response (convolution kernel) which is modified by the range. In fact, conventional techniques for SAR imaging technique can be considered as ray-tracing based SAR imaging technique. It is an efficient technique to obtain focused radar images, when the medium is homogeneous so that the rays of EM wave propagation can be treated as straight lines. However, in case of layered materials the waves are refracted on the interface between two different materials. In contrast, phase shift migration based SAR imaging technique is through EM field extrapolation by phase shift to obtain the reflected EM intensity and radar imaging of scatterers' reflectivity in the radar illumination area. Although EM wave is refracted on the interface of different materials, the EM wave phase is unchanged. Compared to conventional SAR imaging technique, phase shift migration based SAR imaging approach is more suited to obtain focused radar subsurface imaging for GPR systems. In this paper the phase shift migration approach is applied to experimental data as well as practical measurement data. Meanwhile, a background removal algorithm and a spreading and exponential correcting technique is applied to improve the achievement. The presented focused subsurface and scatters radar imaging results show that phase shift migration based SAR technique is satisfied for radar subsurface imaging and inhomogeneous medium imaging.

I. INTRODUCTION Migration imaging technique originates from seismic

imaging [START_REF] Prati | Seismic migration for sar focusing: Interferometrical applications[END_REF]. The first migration approaches are devised to image the seismic wave patterns. Due to the potential to image underground objects the migration algorithms for ground penetrating radar imaging were developed and applied on oil and other source exploration in last decades. These migration techniques are useful for increasing signal to noise ratio of the investigated radar images and improving the azimuth resolution. The application of migration imaging technique on radar EM wave subsurface imaging, e.g. GPR radar imaging, is evolved into the phase shift migration algorithm. The phase shift migration based SAR imaging method is based on the wave equations and its objective is to refocus the reflection signatures in recorded data back to the true positions and obtain shape of the reflection target.

Conventional synthetic aperture radar imaging techniques mainly achieving focused radar imaging is based on the physical-mathematical model that the radar received signal is sum of all reflection signals with corresponding phase delay. To derive the positions of the reflectors and achieve scatterers imaging inversely, as the delay-and-sum, conventional SAR imaging technique summarize the coherent backscattered data after proper phase compensation. Meanwhile, the received signal physical mathematical model can also be formulated simply as the convolution of the reflectors reflectivity function with a range reference signal and an azimuth reference signal. Such as range-doppler-SAR, conventional SAR can also achieve the focused radar imaging through range signal cross correlations (matched filter) and azimuth signal cross correlation (matched filter) with a certain range migration. Whatever the conventional SAR technique is applied, the received signal is formulated as the sum of the reflections signals with corresponding phase delay. When the radar EM wave propagates through the homogeneous medium, the trace of the EM wave transmitted by radar and reflected by scatterer can be seen as straight lines. The phase delay of the reflections signals can be built easily based on the geology relationship between the radar and scatterers. However, the phase delay of the reflections signals are difficult to be built, when the EM wave propagates through inhomogeneous medium and the EM wave tracings are not straight lines. Therefore, conventional SAR technique is feeble on focusing inhomogeneous material radar imaging. In fact the conventional synthetic aperture radar imaging techniques obtain the radar imaging by using signal processing way. However, the phase migration based SAR algorithm is mainly based on wave equation. Through EM field extrapolation, the phase shift migration approach obtains reflectivity function of scatters and achieve the objects radar imaging. Furthermore, phase migration based SAR algorithm has high computation efficiency because it reconstructs images iteratively using the fast Fourier transform (FFT).

In this paper, the phase shift migration based SAR imaging algorithm is discussed and applied on dual polarized GPR imaging. To simplify the discussion, the EM wave propagation loss in medium is ignored and the two dimension phase shift migration approach is described. The configuration schematic diagram of the GPR system (radar subsurface imaging system) is as Fig. [START_REF] Prati | Seismic migration for sar focusing: Interferometrical applications[END_REF]. The radar is on the plane z 0 =0and moves from left side to right side along x-axis. During the moving, radar transmits the EM wave and receives the reflected signal from the illuminated surface, subsurface and buried objects. The coordinate x is the radar aperture dimension (radar moving direction), z is the coordinate in the range dimension. Since we discuss 2-D radar subsurface imaging, the EM field distribution along y-axis is assumed to be constant.

According to the EM wave equation, an EM field u (x, z, t) can be described by the following scalar wave equation:

 ∂ 2 ∂x 2 + ∂ 2 ∂z 2 - 1 v 2 m ∂ 2 ∂t 2 u (x, z, t)=0 (1)
where v m is the electromagnetic wave propagation velocity in the medium, t is the time. According to the property of Fourier transform, applying a three-dimensional Fourier transform on equation ( 1) over x domain, z domain and t domain yields

✓ k 2 x + k 2 z - ω 2 v 2 m ◆ U (k x ,k z ,ω)=0 (2)
where k x , k z are the x direction and z direction components of wave number vector, ω is the angular frequency corresponding to time t. Since U (k x ,k z ,ω) 6 =0, we have

k 2 x + k 2 z - ω 2 v 2 m =0 (3)
Equation ( 3) is known as the dispersion relation for 2-D wave equations. Now applying a two-dimensional Fourier transform on the equation ( 1) over x domain and t domain generates

✓ k 2 x + ∂ 2 ∂z 2 - ω 2 v 2 m ◆ U (k x ,z,ω)=0 (4)
Substituting eq. ( 3) into eq. ( 4), the equation can be obtained as

✓ ∂ 2 ∂z 2 -k 2 z ◆ U (k x ,z,ω)=0 (5)
Considering equation ( 5) to be a homogeneous second-order linear constant coefficient ordinary differential equation, the solution is as

U (k x ,z,ω)=C 0 e -jkzz + C 1 e jkzz (6)
where C 0 , C 1 are constant coefficients. C 0 e -jkzz and C 1 e jkzz are corresponding the upgoing wave and downgoing wave.

Because there is only an upgoing EM wave in the measurement data. Equation ( 6) can be rewritten as:

U (k x ,z,ω)=C 0 e -jkzz (7)
Assigning z =0, we can obtain C 0 = U (k x , 0,ω). Considering eq. ( 3), the solution of eq. ( 5) can be expressed as

U (k x ,z,ω)=U (k x , 0,ω)e -jz q ω 2 v 2 m -k 2 x ( 8
)
where U (k x , 0,ω) is the electromagnetic field in frequency wave number domain at range z =0, where is the radar plane. U (k x , 0,ω) can be obtained by applying two-dimensions Fourier transform on u (x, z =0,t), which again represents the radar measurement data. Because U (k x , 0,ω) can be obtained and known, the electromagnetic field in frequency wave number domain can be determined and is unique based on equation (8). This equation shows that the extrapolation of electromagnetic field U (k x ,z,ω) in wave number-frequency domain along the z-axis is just a phase shift operation. By recursively extrapolating the wave field in wave numberfrequency domain along the z-axis in steps ∆z and using the resulting EM field of each step as input to the next iteration, the frequency-wave number domain distribution of the field U (k x ,z,ω) can be reconstructed. After the field distribution in frequency-wave number domain is reconstructed, the electromagnetic field distribution u (x, z, t) in time-space domain can be calculated by using inverse Fourier transform. This electromagnetic field u (x, z, t) stands for the received data at the range position z. Assigning t =0 , u (x, z, t = 0) means the radar received data on the time equal to zero at the position (x, z), which also means the reflection intensity on the point (x, z). According to above discussion u (x, z, t = 0) is the focused radar image after phase migration procedure. The procedure of the 2-D phase shift migration is as:

Step 1. Fourier transform of the measurement data over x and t:

U (k x ,z 0 =0,ω)= ZZ u (x, z 0 =0,t)e -jkxx e -jωt dxdt
(9) When the radar system uses stepped frequency signal, the received signal is as U (x, z =0,ω), and the Fourier transform operating is only needed over x:

U (k x ,z 0 =0,ω)= ZZ u (x, z 0 =0,ω)e -jkxx dx (10)
Actually, the received signal is recorded along the x direction with limited discrete locations. Instead of Fourier transform (FT), the discrete Fourier transform (DFT) should be used. However, for computational efficiency, fast Fourier transform (FFT) is used in practice.

Step 2. Calculate the electromagnetic field at a new range location z 1 = z 0 +∆ z in frequency wave number domain by phase migration operation as eq. ( 8). Considering the radar signal as round trip signal, the electromagnetic field at a new range location can be expressed as

U (k x ,z 1 ,ω)=U (k x ,z 0 =0,ω)exp(j s 4ω 2 v 2 m -k 2 x ∆z) (11)
Step 3. Using inverse Fourier transform operation along k x direction to obtain the electromagnetic field into frequencyspace domain

u (x, z = z 1 ,ω)= 1 2π Z U (k x ,z = z 1 ,ω)e jkxx dk x (12)
Step 4. Using inverse Fourier transform operation along ω direction to obtain the electromagnetic field into time-space domain and choose time equal to zero t =0to obtain the reflection intensity as local position.

u (x, z = z 1 ,t = 0) = 1 2π Z u (x, z = z 1 ,ω)e ω×(t=0) dω (13)
Step 5. Repeat steps 2 to 4 to obtain the electromagnetic field distribution in frequency-wave number domain and reflection (scattering) intensity image.

The procedure steps of phase migration is as Fig. [START_REF] Jol | Ground Penetrating Radar Theory and Applications[END_REF]. In physical understanding, the procedure of phase migration is like parallel shifting the position of radar observation (measurement) plane from z =0to a position z = z 1 . By phase shift migration processing, the electromagnetic field u (x, z = z 1 ,t) can be calculated by practically measurement u (x, z =0,t), and u(x, z = z 1 ,t) means the equivalent measured data by radar on the location (x, z 1 ). Selecting t =0 value of the equivalent measurement data u(x, z = z 1 ,t) obtains the local reflection, while t =0means no time delay return. Equivalently, phase migration processing shifts the location of radar measurement. When the location of radar measurement is same as the location of targets, the reflection of targets will appear on the measurement time equal to zeros t =0.

It should be noted that it is difficult to have a focused radar subsurface imaging by applying the conventional synthetic aperture radar (SAR) technique on ground penetrating radar system because of the existence of electromagnetic refraction on the subsurface interfaces. When electromagnetic wave penetrated into the subsurface, the propagation velocity will be influenced by the subsurface medium and the propagation direction will also be changed due to the refraction appearing on the interface between two different dielectric constant mediums. In this case, the signal phase delays do not mean the direct distance between the radar and targets. However, phase

III. APPLICATION OF PHASE SHIFT MIGRATION APPROACH

To detect buried objects and obtain the subsurface structure imaging precisely, the dual polarization GPR system is implemented. It operates over the frequency band from 0.3 GHz to 3 GHz with stepped frequency signal. Both of the transmitter and receiver consist of dual polarized double Vivaldi tapered slot antennas. By using the dual polarization antenna as transmitter and receiver, the radar system can simultaneously receive co and cross polarization echo (reflection signal). The measurement is made on a frozen pond. The measurement data and processed radar imaging of the most interesting track in time domain are shown as Fig. [START_REF] Daniels | ser. IEE radar, sonar, navigation and avionics series[END_REF]. Obviously, the phase shift migration approach processed radar imaging is more focused. The first strong return is the reflection by the ice surface. The reflection of pool bed, basalt bed and buried scatterers can also be seen.

A. Background Removal

For GPR systems one important objective is to find buried scatterers. For this aim, the reflection signals from surface and subsurface interfaces should be removed [START_REF] Jol | Ground Penetrating Radar Theory and Applications[END_REF]. However, the reflections of buried objects appears only when the radar Based on the difference of reflections from the buried objects and background during radar moving, a background removal algorithm is implemented on removing the reflections from the surface and subsurface interface. The background removal approach can be expressed as:

û (x i ,z =0,t)=u (x i ,z =0,t) - 1 N x Nx X j=1 u (x j ,z =0,t) (14
) where N x is the measurement times along x-axis.

B. Spreading and exponential correction

The waves transmitted by the GPR are subjected to the free-space path loss and losses in the material. Therefore, the reflections of scatterers in deep range become relatively weak.

To compensate these attenuations and obtain relative accurate scatterers reflections, a spreading and exponential correction (SEC) algorithm is applied on the measurement data [START_REF] Jol | Ground Penetrating Radar Theory and Applications[END_REF], [START_REF] Daniels | ser. IEE radar, sonar, navigation and avionics series[END_REF]. The main idea of the SEC algorithm is increasing (or compensating) the reflection signal amplitude as the reflection signal time delay increasing. The compensation works as a gain function g(t), which is expressed as:

g (t)= 8 < : ⇣ t-t0 t0 ⌘ sc • e t-t 0 t end -t 0 •ec t ≥ t 0 0 t<t 0 ( 15
)
where t 0 is the start point of the SEC technique application, t is the time delay of the reflection signal. t end represents the unique domain of the radar. The sprading correction is adjusted by sc, while by ec the medium losses are compensated. The values for sc and ec were adjusted, so that the phase migration yields the most interpretable results. After applying the background removal and SEC algorithm, the phase shift migration processed radar imaging is as Fig. (4b). Comparing the phase shift migration processed radar imaging without background removal and SEC algorithm as figure(4a), it is obvious to find that the background reflections disappeared and the scatter reflections become more clear.

IV. CONCLUSION

In this paper the phase shift migration based SAR imaging technique are described in detail. In the final paper we'll turn our attention to more measured and processed data, for instance the cross-polarized data.

Fig. 2 .

 2 Fig. 2. Flow chart of phase migration

Fig. 3 .Fig. 4 .

 34 Fig. 3. Phase migration: the red arrows marks two hyperbola in the raw data (left) which are focused in the phase migration processed results (right)


Wolf-Stefan Benedix
email: wolf-stefan.benedix@tu-dresden.de

Dirk Plettemeier
email: dirk.plettemeier@tu-dresden.de

André Zanoni

Fabian Preller

Valerie Ciarletti

Advance of WISDOM GPR Antenna for ExoMars 2018 Mission

The Experiment "Water Ice and Subsurface Deposit Observations on Mars" (WISDOM) is a Ground Penetrating Radar (GPR) selected to be part of the Pasteur payload on board the rover of European Space Agency's (ESA) ExoMars 2018 mission. The GPR antenna system described in this paper is the consequent progression of former developments [1, 2] incorporating changed requirements and further optimizations. Main constraints are the mass, the temperature range as well as the ultra-wide band demand. The antenna requirements which are to fulfill for this very specific GPR application are described here. Furthermore, it is given an overview about the lightweight design and its realization. Simulated and measured antenna performance is compared in this paper.

I. INTRODUCTION

One of the most important goal of the ExoMars2018 mission describes the quest of present or past life forms. One crucial instrument represents the ground penetrating radar experiment WISDOM, with which assistance the geology of the near soil layers will be investigated and furthermore the selection process of suitable drill sites are supported.

This instrument itself has been designed to characterize the shallow subsurface structure of Mars [START_REF] Ciarletti | Wisdom gpr designed for shallow and high-resolution sounding of the martian subsurface[END_REF]. The radar is a gated step frequency system covering a frequency range from 0.5 GHz to 3 GHz [START_REF] Oyan | Ultrawideband gated step frequency groundpenetrating radar[END_REF].

The antenna actually consists of two antennas sending and receiving two orthogonal polarizations each.

II. ANTENNA REQUIREMENTS

The demands result from the application as ground penetrating radar antenna on the one hand and on the other hand from the environmental conditions of the mission. The latter vary over the different phases (ground phase, transportation phase, Mars surface) and show particularly high loads during launch (mechanical loads) and during the surface mission on Mars (thermal loads).

Table I shows the electrical requirements, which arise from intended use while Table II lists the demands from the most critical environmental constraints of the mission.

III. ELECTRICAL PROPERTIES

A. Divider

The antenna design was developed based on a Tapered Slot Antenna with exponential taper (Vivaldi). This kind of antenna fulfills several requirements very well and possesses a comparatively low mass. The construction as a two element array per polarization plane yields in a reduced height of the antenna model. However, this setup needs a broad band power divider with low insertion losses. Several concepts have been evaluated. One of the promising drafts used a transformation divider. But due to the large operating temperature range the use of a divider based on the transformation principle might be critical, since the influence on the magnetic materials is not known. Therefore the so called Wilkinson power divider has been employed. In the former design a five-step variation was developed. But this design occupies a space of about 38 mm × 25 mm and its integration was a demanding step during the assembly of the antenna. For this reason, the divider is minimized 21 mm × 16 mm at a reasonable cost of performance changes [START_REF] Schroeder | Entwicklung einer Teilerstruktur[END_REF]. Figure 1 shows both the old and new power divider design. The diagram of Figure 2 represents measurement results of the former divider. One can see that over the full frequency range the matching of all ports as well as the isolation are better than -20 dB, while the insertion loss is lower than 2 dB. In comparison the values of the simulation results of the new design in Figure 3 (S 11 only) do not reach its predecessor.

B. Optimization of Vivaldi taper

This section will be described in the full paper, since the optimization is not completed yet.

IV. MECHANICAL PROPERTIES

The most limiting factor of the antenna design constitutes the mass budget of 400 g maximum. From a design change of the antenna mount, new requirements for the antenna base plates revealed. With the redesign the both separate antenna are combined on a single base plate. This eventually leads to a mass reduction in several points:

• less screws (from 16 to 9) • less structural side elements • narrow base plate On the other hand the core material of the base plate have to be changed from glass fiber reinforced plastic (GFRP) to aluminum to withstand the mechanical loads. Figure 4 shows an intermediate design study where the sheets of the base plate sandwich structure still made of carbon fiber reinforced plastic (CFRP).

The structure reinforcing elements at the bottom of the antenna are realized as its predecessor by honey comb structure on the basis of a single GFRP material.

By reducing the thickness of the copper layer of 18 µm to 5 µm, the accelerated masses are reduced to the extent that Fig. 5: Geometrical mathematical model of the WISDOM antenna the honeycomb structure with thinner materials withstands the mechanical loads with a safety margin.

V. T HERMAL PROPERTIES

A. Thermal loads

As one of the external survey instruments the WISDOM antenna has to withstand a large temperature range between -120°C and +115°C. The first bound derives from the lowest conceivable temperature on the Mars surface at the expected position. The latter is due to a sterilization requirement with a dry heat microbial reduction. The FEM calculation show, that large temperature changes (from 20°C down to -120°) do not reveal critical stresses on the material.

B. Thermal analysis

The thermal influences of the antenna have been analyzed with a thermal mathematical model of the antenna. The geometrical mathematical model of the intermediate design study is shown in Figure 5.

VI. CONCLUSION -OUTLOOK

In the final paper we will present measurements of the new divider as well as radiation pattern and gain function of the antenna system. Further the results of the thermal analysis are shown. All points will be discussed in detail.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Comparison of the a) former and b) recent version of the Wilkinson divider (not to scale)

Fig. 3 :Fig. 4 :

 34 Fig. 3: Measurement results of the recent Wilkinson divider: S 11 -> brown, other colors represent different divider designs

TABLE I :

 I List of electrical requirements of the WISDOM antenna

	Requirement	Value
	Frequency bandwidth	0.5...3.0 GHz = 2.5 GHz
	Polarization (Tx, Rx)	linear, XX -XY -YX -YY
	Matching (S 11 )	≤-10 dB
	Radiation pattern	homogenous main lobe
	Beam width	> 20°, H-plane > E-plane
	Gain function	smooth, increasing with frequency
	Antenna cross talk	< -15 dB (for all polarizations)
	Pointing error	< 5°(for all frequencies)
	Radiation coupling to rover	as low as possible

TABLE II :

 II List of mechanical and thermal requirements of the WISDOM antenna

	Requirement	Value
	First mechanical resonance frequency	140 Hz
	Mass budget	400 g
	Envelope	200 mm × 410 mm × 180 mm
	Temperature range (non-op)	-120°C to +115°C
	Temperature range (operating)	-90°C to +115°C

ACKNOWLEDGMENT WISDOM development and this research were supported by funding from the French space agency CNES and from the German space agency DLR.


E L Houcein

MARIUSZ LEMA ŃCZYK E L Abdalaoui

Thierry De

L A Rue

A DYNAMICAL POINT OF VIEW ON THE SET OF B-FREE INTEGERS

Sarnak has recently initiated the study of the Möbius function and its square, the characteristic function of square-free integers, from a dynamical point of view, introducing the Möbius flow and the square-free flow as the action of the shift map on the respective subshfits generated by these functions.

In this paper, we extend the study of the square-free flow to the more general context of B-free integers, that is to say integers with no factor in a given family B of pairwise relatively prime integers, the sum of whose reciprocals is finite. Relying on dynamical arguments, we prove in particular that the distribution of patterns in the characteristic function of the B-free integers follows a shift-invariant probability measure, and gives rise to a measurable dynamical system isomorphic to a specific minimal rotation on a compact group. As a by-product, we get the abundance of twin B-free numbers. Moreover, we show that the distribution of patterns in small intervals of the form [N, N + √ N ) also conforms to the same measure.

When elements of B are squares, we introduce a generalization of the Möbius function, and discuss a conjecture of Chowla in this broader context.

Introduction and overview of the paper

Throughout the paper, the symbol N * denotes the set of natural numbers starting from 1: N * := {1, 2, 3, . . .}. For any set A, we denote by |A| the cardinality of A.

We are interested in arithmetic functions defined on N * and taking values in a subset Γ ⊂ {-1, 0, 1}. We view these functions as points x = (x n ) n ∈ N * in the compact topological space Γ N * . On this space, we consider the shift map S : x = (x n ) n∈N * → Sx = (x ′ n ) n∈N * , where for each n ∈ N * , x ′ n := x n+1 . We say that x ∈ Γ N * is generic for some probability measure ν on Γ N * if the frequency of patterns in the sequence x conforms to ν. Formally, this means that for each cylinder set C ⊂ Γ N * , we have 1

N 0≤n<N ½ C (S n x) ----→ N →∞ ν(C).
Observe that if x is generic for ν, then ν must be shift-invariant.

1.1. The Möbius flow and the square-free flow. Recall that the Möbius function is the multiplicative arithmetic function µ defined, for all positive integer n, by µ(n) := 0 if n is divisible by the square of a prime number, (-1) dn otherwise, where d n is the number of prime factors of n.

Research supported by Science National Center (NCN) grant DEC-2011/03/B/ST1/00407. The results of the paper have been obtained during the visit of the second author at Rouen University in September 2013.

Its square µ 2 is the characteristic function of the square-free integers. In spite of their seemingly artificial definitions, these functions are of great importance in number theory for their connections with the distribution of prime numbers and the Riemann zeta function (see e.g. [START_REF] Tom | Introduction to analytic number theory[END_REF][START_REF] Hildebrand | Introduction to analytic number theory[END_REF][START_REF] Titchmarsh | The theory of the Riemann zeta-function[END_REF]).

Recently, Sarnak [START_REF] Sarnak | Three lectures on the Möbius function randomness and dynamics[END_REF] has suggested to study µ and µ 2 from a dynamical point of view: He introduced the Möbius flow (respectively the square-free flow ) as the action of the shift map on the closure of the orbit of µ in {-1, 0, 1} N * (respectively of µ 2 in {0, 1} N * ). Sarnak announced several results concerning the square-free flow: Its topological entropy is 6/π 2 (when the logarithm is taken in base 2), and the sequence µ 2 is generic for a shift-invariant probability measure ν S on {0, 1} N * which has zero Kolmogorov entropy. He also stated important conjectures on the Möbius flow: In particular, he explained that the sequence µ is expected to be generic for a probability measure ν N which can be seen as the "fully random" extension of ν S , and connects this conjecture to a famous conjecture of Chowla [START_REF] Chowla | The Riemann hypothesis and Hilbert's tenth problem[END_REF].

Our purpose in the present work is to extend this point of view to a generalized setting, providing simple dynamical proofs of several known results concerning square-free numbers, and extending them to the context of so-called B-free numbers (see below). We also consider a generalization of the Möbius function, which we hope can shed a new light upon the Chowla conjecture.

k≥1 1 b k < ∞. (2)
Integers with no factors in B are called B-free. Those integers have been introduced by Erdös [START_REF] Erdős | On the difference of consecutive terms of sequences defined by divisibility properties[END_REF], who proved the existence of some 0 < c < 1 such that, for all large enough N , the interval [N, N + N c ) contains at least one B-free number. A special case of B-free numbers, in which falls the case of square-free numbers, had been previously considered by Mirsky [START_REF] Mirsky | Arithmetical pattern problems relating to divisibility by rth powers[END_REF], who studied the distribution of patterns in the characteristic function of r-free numbers, that is to say numbers which are not divisible by the r-th power of any prime (r ≥ 2). We introduce the sequence η = (η n ) n∈N * which is the characteristic function of the set of B-free integers:

(3)

η n := 0 if there exists k ≥ 1 such that b k divides n, 1 otherwise, i.e. if n is B-free.
We define a subshift X B ⊂ {0, 1} N * (Section 2.2), and a shift-invariant probability measure ν B on X B (Section 2.3) such that:

• The closure of the orbit of η under the shift map is X B (Corollary 4.2).

• The topological entropy of the subshift X B is positive, equal to

∞ k=1 (1 - 1/b k ) (Theorem 5.
3). This generalizes the formula given by Sarnak [START_REF] Sarnak | Three lectures on the Möbius function randomness and dynamics[END_REF] in the context of the square-free flow, a proof of which can be found in [START_REF] Peckner | Uniqueness of the measure of maximal entropy for the squarefree flow[END_REF].

• The sequence η is ν B -generic (Theorem 4.1).

• The dynamical system (X B , ν B , S) has zero Kolmogorov entropy, even discrete spectrum. In fact, we provide an explicit measurable isomorphism between this system and the minimal rotation T , which is the addition of (1, 1, . . .) on the compact abelian group Ω := k≥1 Z/b k Z (Theorem 3.1).

The last point generalizes the main result of Cellarosi and Sinai [START_REF] Cellarosi | Ergodic properties of square-free numbers[END_REF], concerning the square-free flow, to the context of B-free numbers. But we point out that, while these authors make use of Mirsky's theorem providing the frequency of patterns in the sequence µ 2 , our Theorem 4.1 provides an alternative, purely dynamic, proof of Mirsky's result. We also show that the method to prove Theorem 4.1 can be adapted to study the frequency of patterns in η along arithmetic subsequences (Theorem 4.4). Moreover, provided an extra condition on B which is automatically satisfied in the case of r-free numbers (r ≥ 2), we prove that the frequency of patterns in η in "short intervals" of the form [N, N + √ N ) is also given by the measure ν B (Theorem 4.5). The existence of B-free numbers in intervals of the form [N, N + N c ) has been very much studied since the work of Erdös [START_REF] Erdős | On the difference of consecutive terms of sequences defined by divisibility properties[END_REF], in which it is conjectured that for any c > 0, the interval [N, N + N c ) always contains at least one B-free number for N large enough (see [START_REF] Wu | Distribution des nombres B-libres dans de petits intervalles[END_REF], [START_REF] Alkan | A survey on the distribution of B-free numbers[END_REF] and references therein). But, to our knowledge, this is the first time that frequency of patterns inside short intervals is considered.

1.3.

A generalization of the Möbius function. In the last section of the present paper, we propose to study a generalization of the Möbius function. We assume here that all integers b k ∈ B are squares: b k = a 2 k , where the integers a k are pairwise relatively prime. For each positive integer n, we define δ n as the number of integers a k dividing n, and consider the sequence π = (π n ) n∈N * defined by π n := (-1) δn . Finally, we define µ = (µ n ) n∈N * , where µ n := η n • π n . Observe that, if the set {a k : k ≥ 1} is the set of prime numbers, we recover in this way the Möbius function: µ n = µ(n). (We therefore refer to this situation as the Möbius case.) We introduce an abstract property of the sequence µ, called the Chowla property, which in the Möbius case corresponds exactly to the Chowla conjecture. We show that this Chowla property is equivalent to the genericity of µ for a shift-invariant probability measure ν M on {-1, 0, 1} N * , which is the "completely random" extension of ν B (Theorem 6.1). Moreover, this property is satisfied as soon as π is generic for the Bernoulli measure

β := (1/2, 1/2) ⊗N * on {-1, 1} N * .
An important parameter in our study is Σ := k≥1 1/a k . When Σ < ∞, we prove that π is indeed generic, however the shift-invariant measure ν ′ which is obtained has zero Kolmogorov entropy (Theorem 6.2). In the Möbius case, we have Σ = ∞, and we are not able to say whether π is generic for some measure or not. However, we show that, as Σ → ∞, the probability measure ν ′ for which π is generic converges weakly to the Bernoulli measure β (Theorem 6.4).

The B-free flow

In this section, we introduce the study of the sequence η, which is the characteristic function of the set of B-free integers, from a dynamical point of view. We work here in the space X := {0, 1} N * . 2.1. A dynamical system which outputs η. The starting point of our argument is the following simple fact: We can observe η along the orbit of a particular point in a specific dynamical system. Indeed, consider the compact additive group

Ω := k≥1 Z/b k Z,
and the transformation T of Ω mapping ω = (ω k ) k≥1 to T ω := (ω k + 1) k≥1 . Then we can get our sequence η by observing the function f : Ω → {0, 1} defined by f (ω) := 0 if there exists k ≥ 1 such that ω k = 0, 1 otherwise.

along the orbit of the element ω := (0, 0, 0, . . .) ∈ Ω:

(4) ∀n ∈ N * , η n = f (T n ω).
More generally, it will be useful to introduce the map ϕ : Ω → X defined by ( 5)

ϕ(ω) := f (T n ω) n∈N * , so that η = ϕ(ω). Note that ϕ(ω) is the sequence (x n ) n∈N * ∈ X satisfying (6) ∀n ∈ N * , x n = 0 if ∃k ≥ 1 : ω k + n = 0 [b k ], 1 if ∀k ≥ 1 : ω k + n = 0 [b k ].
By construction, we have Definition 2.1. A subset A ⊂ N * is said to be B-admissible if

S • ϕ = ϕ • T (indeed, (Sϕ(ω)) n = ϕ(ω) n+1 = (ϕ(T ω)) n ). However, ϕ is not a topological factor map because it is not continuous (e.g., if 2 / ∈ B, ϕ(1, 1, . . . , 1, 1, . . .) 1 = 1 = 0 = ϕ(1, 1, . . . , 1, -1, -1, . . .) 1 ).
(7) ∀k ≥ 1, t(A, b k ) < b k .
An infinite sequence x = (x n ) n∈N * ∈ X is said to be B-admissible if its support {n ∈ N * : x n = 1} is B-admissible. In the same way, a finite block

x 1 . . . x N ∈ {0, 1} N is B-admissible if {n ∈ {1, . . . , N } : x n = 1} is B-admissible.
We denote by X B the set of all B-admissible sequences in X. Since a set is Badmissible if and only if each of its finite subsets is B-admissible, and a translation of a B-admissible set is B-admissible, X B is a closed and shift-invariant subset of X, i.e. a subshift. Proposition 2.2. For any ω ∈ Ω, ϕ(ω) is a B-admissible sequence. In particular, η is B-admissible.

Proof. Let ω ∈ Ω, and denote by A the support of ϕ(ω). Then for each k ≥ 1, we have -ω k ∈ {n mod b k : n ∈ A}.

2.3.

A shift-invariant measure on X B . Since the integers b k are pairwise relatively prime, the topological dynamical system (Ω, T ) is a minimal rotation, hence also uniquely ergodic, and its unique ergodic invariant probability measure is the normalized Haar measure on Ω, which we denote by P. Under P, the coordinates ω k , k ≥ 1 are independent, each of them being uniformly distributed in the corresponding finite group Z/b k Z.

Let ν B be the pushforward measure of P to X by the map ϕ: For each measurable subset C ⊂ X, ν B (C) := P(ϕ -1 C). Then ν B is shift-invariant, and by Proposition 2.2, ν B is concentrated on X B . Moreover, the measurable dynamical system (X B , ν B , S) is a factor of (Ω, P, T ). In particular, it is ergodic.

We shall need in the sequel the following lemma, valid in the context of any probability space (Ω, P).

Lemma 2.3. In the probability space (Ω, P), let (E n ) n∈N * be a sequence of events, and for each n, let F n := Ω \ E n . Then, for any finite disjoint subsets A, B ⊂ N * , we have

P n∈A E n ∩ m∈B F m = A⊂D⊂A∪B (-1) |D\A| P d∈D E d .
Proof. Denoting by E[ • ] the expectation with respect to the probability P, we have

P n∈A E n ∩ m∈B F m = E n∈A ½ En m∈B (1 -½ Em ) = C⊂B (-1) |C| E n∈A∪C ½ En = A⊂D⊂A∪B (-1) |D\A| P d∈D E d .
Now we want to compute the values taken by ν B on cylinder sets in X. For any finite disjoint subsets A, B ⊂ N * , we denote by C A,B the cylinder set

C A,B := {x = (x n ) n∈N * ∈ X : ∀n ∈ A, x n = 1, ∀m ∈ B, x m = 0} .
When B is empty, we rather write C 1 A for C A,∅ . Analogously, we write C 0 B for C ∅,B . Proposition 2.4. For any finite disjoint subsets A, B ⊂ N * , we have

(8) ν B (C 1 A ) = k≥1 1 - t(A, b k ) b k , and
(9) ν B (C A,B ) = A⊂D⊂A∪B (-1) |D\A| k≥1 1 - t(D, b k ) b k .
Proof. Let A be a finite subset of N * . First notice that for a fixed k ≥ 1,

P ω ∈ Ω : ∀n ∈ A, ω k + n = 0 [b k ] = 1 - t(A, b k ) b k .
Then, we have

ν B (C 1 A ) = P ϕ -1 (C 1 A ) = P   k≥1 ω ∈ Ω : ∀n ∈ A, ω k + n = 0 [b k ]   = k≥1 1 - t(A, b k ) b k ,
which proves [START_REF] Heath-Brown | The square sieve and consecutive square-free numbers[END_REF]. Now, observe that ( 9) can be derived from ( 8) using Lemma 2.3.

Proposition 2.5. For any finite disjoint subsets A, B ⊂ N * , the following properties are equivalent:

(i) ν B (C A,B ) > 0; (ii) ν B (C 1 A ) > 0; (iii) A is a B-admissible set.
Proof. By ( 8) and ( 2), (ii) is equivalent to t(A, b k ) < b k for all k ≥ 1, which is the definition of a B-admissible set. Obviously, (i) implies (ii), and it only remains to show that (ii)=⇒(i).

Assume that

ν B (C 1 A ) > 0. We want to prove that ν B (C A,B ) = P(ϕ -1 C A,B ) > 0. Since A and B are disjoint, and since b k ----→ k→∞ ∞ by (2), there exists K such that, for all j ≥ 1, the equation n = m [b K+j ] has no solution with n ∈ A and m ∈ B. Let B = {m 1 , . . . , m r }. Observe that ω ∈ Ω : ∀1 ≤ j ≤ r, ω K+j + m j = 0 [b K+j ] ∩ ω ∈ Ω : ∀k / ∈ {K + 1, . . . , K + r}, ∀n ∈ A, ω k + n = 0 [b k ] ⊂ ϕ -1 (C A,B ).
But the set on the left-hand side is the intersection of two independent events in (Ω, P), the first one has probability r j=1 1/b K+j > 0, and the second one contains ϕ -1 (C 1 A ), therefore has positive probability. Corollary 2.6. The support of ν B is the subshift X B of B-admissible sequences.

The measurable dynamical system (X B , ν B , S)

The purpose of this section is to prove the following result, which provides a complete description of the measurable dynamical system (X B , ν B , S).

Theorem 3.1. The map ϕ is an isomorphism between the two measurable dynamical systems (Ω, P, T ) and (X B , ν B , S).

We already know that ϕ is a measurable factor map from (Ω, P, T ) to (X B , ν B , S). Therefore the proof of the theorem reduces to the proof of the following result. Proposition 3.2. There exists Ω 0 ⊂ Ω, with P(Ω 0 ) = 1, such that for each ω ∈ Ω 0 , the preimage of ϕ(ω) is reduced to the singleton {ω}.

z ∈ Z/b k Z the subset Ω k,z := {ω ∈ Ω : ω k = z}. Obviously Ω k,z is invariant under the action of T b k . We also note that, for all k ′ = k, since the integer b k ′ is relatively prime with b k , the addition of b k in Z/b k ′ Z is isomorphic to the addition of 1. Let Ω := k ′ =k ′ Z/b k ′ Z, denote
by T the addition of (1, 1, . . .) in Ω, and by P the normalized Haar measure on Ω. Then the dynamical system Ω k,z , P( • | Ω k,z ), T b k is measurably isomorphic to ( Ω, P, T ). Since the latter is an ergodic rotation, the action of

T b k on Ω k,z is ergodic. Now, if A ⊂ Ω is a T b k -invariant set, by ergodicity, we must have P(A | Ω k,z ) ∈ {0, 1}. Therefore A coincides modulo P with the union of those Ω k,z satisfying P(A | Ω k,z ) = 1. Proof of Proposition 3.2. Fix k ≥ 1 and z ∈ Z/b k Z. Since ϕ • T b k = S b k • ϕ, for any z ∈ Z/b k Z, the event E k,z := {ω ∈ Ω : ϕ(ω) = 0 along π z,b k } is T b k -invariant,
and contains Ω k,z . Hence, by Lemma 3.3, it coincides P-almost surely with an ω k -measurable event, which is of the form ω ∈ Ω : ω k ∈ {z 1 , . . . , z ℓ } for some finite subset {z 1 , . . . , z ℓ } ⊂ Z/b k Z. We want to show that this finite subset is reduced to the singleton {z}. For this, it is enough to prove that, for any [START_REF] Heath-Brown | The square sieve and consecutive square-free numbers[END_REF]. It follows that

z ′ ∈ Z/b k Z \ {z}, P(E k,z |Ω k,z ′ ) = P(E k,z |ω k = z ′ ) < 1. Fix some n ∈ π z,b k , and note that z ′ + n = 0 [b k ]. We have P(E k,z |ω k = z ′ ) ≤ P ϕ(ω) n = 0 ω k = z ′ = P ∃k ′ = k : ω k ′ + n = 0 [b k ′ ] ω k = z ′ = P ∃k ′ = k : ω k ′ + n = 0 [b k ′ ] by independence in (Ω, P) ≤ P ϕ(ω) n = 0 = 1 -ν B (C 1 {n} ) < 1 by
P E k,z \ {ω ∈ Ω : ω k = z} = 0.
We now get the proposition by setting

Ω 0 := Ω \ k≥1 z∈Z/b k Z E k,z \ {ω ∈ Ω : ω k = z} . Indeed, notice that for each ω ∈ Ω and each k ≥ 1, we have ω ∈ E k,ω k . Assume now that φ(ω ′ ) = φ(ω ′′ ), but that ω ′ = ω ′′ . Then there exists k ≥ 1 with ω ′ k = ω ′′ k . We have ω ′′ ∈ E k,ω ′′ k , but also ω ′ ∈ E k,ω ′′ k since φ(ω ′ ) = φ(ω ′′ ). Thus ω ′ ∈ E k,ω ′′ k \ {ω : ω k = ω ′′ k } and therefore ω ′ / ∈ Ω 0 .

Frequency of blocks in η

We already know by Proposition 2.2 that all finite blocks of 0-s and 1-s appearing in the sequence η have to be B-admissible. The following theorem shows that, conversely, any B-admissible finite block does appear in η, and this with a positive asymptotic frequency given by the probability ν B of the corresponding cylinder.

Theorem 4.1. The sequence η is generic for the probability ν B , i.e. for any cylinder set C ⊂ X, we have [START_REF] Mirsky | Arithmetical pattern problems relating to divisibility by rth powers[END_REF] 1

N 0≤n<N ½ C (S n η) ----→ N →∞ ν B (C).
Proof. Recalling that η = ϕ(ω), we first observe that ( 10) is equivalent to

(11) 1 N 0≤n<N ½ ϕ -1 (C) (T n ω) ----→ N →∞ P ϕ -1 (C) .
It is well known that, under the action of a minimal rotation on a compact group, each element of the group is generic for the normalized Haar measure on the group. So, if ϕ was continuous, (11) would be immediate. Unfortunately ϕ is not continuous, and we have to work a little harder to prove the claim. We note that it is enough to prove [START_REF] Peckner | Uniqueness of the measure of maximal entropy for the squarefree flow[END_REF] in the case where the cylinder set C is of the form C 0 B for some finite subset B ⊂ N * . (Recall that C 0 B is the set of sequences (x n ) ∈ X satisfying x n = 0 for each n ∈ B.) Indeed, if [START_REF] Peckner | Uniqueness of the measure of maximal entropy for the squarefree flow[END_REF] is valid for each such C 0 B , then applying Lemma 2.3 with the family of events E n := {ω : ϕ(ω) n = 0}, both for P and for the empirical measure 1 N 0≤n<N δ T n ω , we can prove that (11) holds for any cylinder set C A,B .

Let B be a finite subset of N * . We observe that [START_REF] Peckner | Uniqueness of the measure of maximal entropy for the squarefree flow[END_REF] for the cylinder set

C 0 B means (12) 1 N 0≤n<N ½ ϕ -1 (C 0 B ) (T n ω) ----→ N →∞ P ϕ -1 (C 0 B ) .
We can write ϕ -1 (C 0 B ) as the union of the increasing sequence of events H ℓ , where

H ℓ := ω ∈ Ω : ∀m ∈ B, ∃1 ≤ k m ≤ ℓ, ω km + m = 0 [b km ] .
We have P(H ℓ ) ---→ ℓ→∞ P ϕ -1 (C 0 B ) . Since ½ H ℓ (ω) only depends on the coordinates ω k , 1 ≤ k ≤ ℓ, and the action of T on the finite group 1≤k≤ℓ Z/b k Z being, by ergodicity of T , a cyclic permutation of the elements of this group, we have [START_REF] Titchmarsh | The theory of the Riemann zeta-function[END_REF] 1

N 0≤n<N ½ H ℓ (T n ω) ----→ N →∞ P(H ℓ ).
To prove [START_REF] Sarnak | Three lectures on the Möbius function randomness and dynamics[END_REF], it only remains to show that [START_REF] Wu | Distribution des nombres B-libres dans de petits intervalles[END_REF] sup

N ≥1 1 N 0≤n<N ½ ϕ -1 (C 0 B )\H ℓ (T n ω) ---→ ℓ→∞ 0. If ω ∈ ϕ -1 (C 0 B ) \ H ℓ , then there exist m ∈ B and k > ℓ such that ω k + m = 0 [b k ]. Therefore we can write (15) ½ ϕ -1 (C 0 B )\H ℓ ≤ m∈B k>ℓ f m,k ,
where

f m,k (ω) := 1 if ω k + m = 0 [b k ], 0 otherwise.
For a fixed m ∈ B and k > ℓ, let us consider the sequence f m,k (T n ω) n∈N * . This is a sequence of 0-s and 1-s, in which two occurences of 1 are separated by a multiple of b k . Moreover, as ω k = 0, the sequence shows no 1 in the interval 1 ≤ n < b k -sup B. It follows that, provided k is large enough so that b k > 2 sup B (which is guaranted if ℓ is large enough), for any N ≥ 1 we have the inequality

(16) 1 N 0≤n<N f m,k (T n ω) ≤ 2 b k .
We then have, for ℓ large enough, 1

N 0≤n<N ½ ϕ -1 (C 0 B )\H ℓ (T n ω) ≤ 2|B| k>ℓ 1 b k ,
which proves [START_REF] Wu | Distribution des nombres B-libres dans de petits intervalles[END_REF]. Applying [START_REF] Mirsky | Arithmetical pattern problems relating to divisibility by rth powers[END_REF] with the cylinder sets

C 1 A when A = {x ∈ X : x 1 = x 3 = 1} or A = {x ∈ X : x 1 = x 2 = 1}
gives also the following interesting consequence of Theorem 4.1.

Corollary 4.3 (Twin B-free integers).

There always exist infinitely many integers n such that n and n + 2 are simultaneously B-free. If 2 / ∈ B, there exist infinitely many integers n such that n and n + 1 are simultaneously B-free.

This result was already known in the context of r-free integers by Mirsky's theorem [START_REF] Mirsky | Arithmetical pattern problems relating to divisibility by rth powers[END_REF], and the question of the speed of convergence of the density has been much studied using the circle method (see e.g. [START_REF] Heath-Brown | The square sieve and consecutive square-free numbers[END_REF][START_REF] Dietmann | The density of twins of k-free numbers[END_REF] and references therein). But to our knowledge its extension to the context of B-free numbers is new. It would be natural to ask whether our method can be adapted to provide a speed of convergence.

4.1.

Averaging along arithmetic subsequences. The method used to prove Theorem 4.1 can also be useful if we are interested in the frequency of blocks in η along arithmetic subsequences. Of course, all admissible blocks do not appear along any arithmetic subsequence since, for instance, η nb1 = 0 for each n ≥ 0. But, as an example of what we can say in this setting, we prove the following result. Theorem 4.4. Let p be a prime number. Then, there exists an integer m ≥ 1, depending only on p and B, such that for any integer s ≥ 1, we have for each

cylinder set C ⊂ X (17) 1 p m p m -1 r=0 1 N 0≤n<N ½ C (S np s +r η) ----→ N →∞ ν B (C).
Proof. We follow the same argument as in the proof of Theorem 4.1, with T p s in place of T . For each b k which is not divisible by p, the addition of p s in Z/b k Z is isomorphic to the addition of 1. The only modification in the argument happens if there exists some b k0 which is divisible by p (in this case, b k0 is unique by ( 1)).

Then, let m be the highest power of p dividing b k0 , define m ′ := min{m, s} and write b k0 = qp m ′ . The Euclidean division by p m ′ of an element j ∈ Z/b k Z (identified with {0, . . . , b k -1}) provides a natural isomorphism between Z/b k Z and the direct product Z/qZ × Z/p m ′ Z. Besides, the addition of p s in Z/b k0 Z is isomorphic to the addition of (1, 0) in Z/qZ × Z/p m ′ Z. Now everything works as in the preceding proof, except that we have added a supplementary cyclic group Z/p m ′ Z on which T p s acts as the identity. Therefore, assuming ℓ ≥ k 0 , (13) becomes

1 p m ′ p m ′ -1 r=0 1 N 0≤n<N ½ H ℓ (T np s +r ω) ----→ N →∞ P(H ℓ ).
Since m ′ ≤ m, p m is a multiple of p m ′ and we also have

1 p m p m -1 r=0 1 N 0≤n<N ½ H ℓ (T np s +r ω) ----→ N →∞ P(H ℓ ).
The end of the proof goes as in the proof of Theorem 4.1.

4.2.

Distribution of B-free numbers in short intervals. We want to show here that the proof of Theorem 4.1 can also be adapted to the case where averaging is done along intervals of the form [N, N + √ N ). The following theorem will show that the frequency of blocks in η in these "short intervals" also conforms to the measure ν B , provided we assume a further hypothesis on the set B: We suppose here that

(18) ∃M, ∀x large enough, B ∩ [x, x + √ x] ≤ M.
Note that the above assumption holds if, for example, there exists an integer r ≥ 2 such that B = {p r : p prime}. Indeed, the length of the interval x 1/r , (x + √ x) 1/r goes to 0 as x → ∞, hence this interval contains at most one prime number for x large. Theorem 4.5. Assume that (18) holds. Then, for any cylinder set C ⊂ X, we have

(19) 1 √ N N ≤n<N + √ N ½ C (S n η) ----→ N →∞ ν B (C).
Proof. We will follow the same lines as in the proof of Theorem 4.1. Again, it is enough to prove that, if C is a cylinder set of the form C 0 B , then the following convergence holds:

(20) 1 √ N N ≤n<N + √ N ½ ϕ -1 (C 0 B ) (T n ω) ----→ N →∞ P ϕ -1 (C 0 B ) .
For ℓ ≥ 1, let H ℓ be defined as in the proof of Theorem 4.1. For the same reason as in [START_REF] Titchmarsh | The theory of the Riemann zeta-function[END_REF], we also have

1 √ N N ≤n<N + √ N ½ H ℓ (T n ω) ----→ N →∞ P(H ℓ ),
and it only remains to show that (21) lim sup

N →∞ 1 √ N N ≤n<N + √ N ½ ϕ -1 (C 0 B )\H ℓ (T n ω) ---→ ℓ→∞ 0.
We can use again (15). Observing that

f m,k (T n ω) = 1 if b k divides n + m, 0 otherwise,
we get that for any fixed N ≥ 1,

(22) 1 √ N N ≤n<N + √ N ½ ϕ -1 (C 0 B )\H ℓ (T n ω) ≤ m∈B k>ℓ 1 √ N N ≤n<N + √ N ½ n+m=0 [b k ] ,
where ½ n+m=0 [b k ] has to be seen as a function of n which is either equal to 1 if

n + m = 0 [b k ],
or to 0 otherwise. Let L ≥ 2 be a large enough number, to be precised later. For a fixed N , we distinguish two kinds of integers b k :

• We say that b k is small if b k ≤ L √ N . • We say that b k is large if b k > L √ N .
Let us consider first the contribution of large b k -s. For such a b k , for any m ∈ B, there exists at most one n ∈

[N, N + √ N ) such that n + m = 0 [b k ]. It follows that N ≤n<N + √ N ½ n+m=0 [b k ] ∈ {0, 1},
and the above expression takes value 1 if and only if there exists N ≤ n < N + √ N such that n + m is a multiple of b k . We rewrite the latter condition as follows: There exists an integer j ≥ 1 with

jb k ∈ [N + m, N + m + √ N ). But, since b k is large, this implies L √ N < b k < N + m + √ N j ,
hence, for N large enough,

j < 1 L √ N + m √ N + 1 < 2 √ N L .
Since √ N /j ≤ (N + m)/j, it follows by assumption (18) that ( 23)

k>ℓ b k large 1 √ N N ≤n<N + √ N ½ n+m=0 [b k ] ≤ 1 √ N 1≤j<2 √ N /L B ∩ (N + m)/j, (N + m + √ N )/j ≤ 2M L ,
hence we can assume that the contribution of large b k -s in the right-hand side of ( 22) is arbitrarily small by choosing L large enough.

Then consider the case of a small b k . If b k > √ N , again for each m ∈ B there exists at most one n ∈ [N, N + √ N ) such that n + m = 0 [b k ], and we have

1 √ N N ≤n<N + √ N ½ n+m=0 [b k ] ≤ 1 √ N ≤ L b k .
On the other hand, if √ N ≥ b k , since different integers n satisfying n + m = 0 [b k ] are separated by multiples of b k , we have as in ( 16)

1 √ N N ≤n<N + √ N ½ n+m=0 [b k ] ≤ 2 b k ≤ L b k .
We finally get, for any m ∈ B,

k>ℓ b k small 1 √ N N ≤n<N + √ N ½ n+m=0 [b k ] ≤ L k>ℓ 1 b k ,
which can be made as small as desired by choosing ℓ large enough. This demonstrates the validity of ( 21), and concludes the proof of the theorem.

One can observe that, as in Corollary 4.3, the previous theorem proves the abundance of twin B-free numbers in short intervals, provided that condition (18) is satisfied.

Entropy of the subshift X B

The purpose of this section is to compute the topological entropy of the subshift X B generated by η, which measures the exponential growth of the number of subwords of length n in η. We denote by γ(n) this number:

γ(n) := {W ∈ {0, 1} n : W is B-admissible} .
For each K ≥ 1, let B K := {b 1 , . . . , b K } ⊂ B. We define B K -admissibility in the same way as B-admissibility, restricting condition [START_REF] Furstenberg | Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation[END_REF] to k ∈ {1, . . . , K}. We then set γ K (n) := {W ∈ {0, 1} n : W is B K -admissible} . Since B-admissibility obviously implies B K -admissibility for all K ≥ 1, we have

(24) ∀K ≥ 1, γ(n) ≤ γ K (n).
For a fixed n, the set of B K -admissible words of length n decreases, as K → ∞, to the set of B-admissible words of length n, so that there exists

K(n) satisfying (25) γ(n) = γ K(n) (n).
We want now to estimate the quantity γ K (n

) for K ≥ 1. For z k ∈ Z/b k Z, 1 ≤ k ≤ K, set Z(z 1 , . . . , z K ) := {n ∈ N * : ∃1 ≤ k ≤ K, n = z k [b k ]} Notice that the set {1, ..., n} \ Z(z 1 , ..., z K ) is B K -admissible. Lemma 5.1. If n is a multiple of b 1 • • • b K , 1 n {1, . . . , n} \ Z(z 1 , . . . , z K ) = K k=1 1 - 1 b k . Proof. If A is a set of b 1 • • • b K consecutive integers, the Chinese Remainder Theo- rem tells us that the map θ : A → K k=1 Z/b k Z defined by θ(j) := j [b 1 ], . . . , j [b K ] is bijective. But θ A \ Z(z 1 , . . . , z K ) is the set of w = (w 1 , . . . , w K ) ∈ K k=1 Z/b k Z satisfying ∀1 ≤ k ≤ K, w k = z k .
Since the cardinality of the latter set is K k=1 (b k -1), we get

|A \ Z(z 1 , . . . , z K )| |A| = K k=1 1 - 1 b k ,
and the lemma follows.

Proposition 5.2. Assume that n is a multiple of b 1 • • • b K . Then (26) 2 n K k=1 1-1 b k ≤ γ K (n) ≤ 2 n K k=1 1-1 b k K k=1 b k .
Proof. Here is a way to produce a B K -admissible word W = w 1 . . . w n of length n:

Step 1. Choose (z 1 , . . . , z K ) ∈ K k=1 Z/b k Z, and set w j = 0 for j ∈ Z(z 1 , . . . , z K ); Step 2. Complete the word by choosing arbitrarily w j ∈ {0, 1} for each j ∈ {1, . . . , n}\ Z(z 1 , . . . , z K ).

Observe that, if we have fixed (z 1 , . . . , z K ) ∈ Theorem 5.3. The topological entropy of the subshift X B is given by

h top (X B ) = ∞ k=1 1 - 1 b k Proof. We have to prove that (27) lim n→∞ 1 n log 2 γ(n) = ∞ k=1 1 - 1 b k . Let K ≥ 1. For each n which is a multiple of the product b 1 • • • b K , we have 1 n log 2 γ(n) ≤ 1 n log 2 γ K (n) (by 24) ≤ K k=1 1 - 1 b k + 1 n log 2 K k=1 b k (by Proposition 5.2).
Of course, if (a k ) is the whole sequence of prime numbers, then µ coincides with the Möbius function. We shall refer to this situation as the Möbius case. In this case, observe that π is a multiplicative arithmetic function, which presents some analogy with the classical Liouville function (but the latter counts the number of prime divisors with multiplicity, which is not the case of our sequence π).

We now need to work in the spaces Y := {-1, 1} N * , and Z := {-1, 0, 1} N * . If the context requires to clarify on which space, X, Y , or Z, acts the shift map, we shall use respectively the notations S X , S Y and S Z .

6.1. An interpretation of the Chowla conjecture. We say that the sequence (µ n ) defined above satisfies the Chowla property if, for any r ≥ 1, any natural numbers 0 ≤ s 1 < . . . < s r and i 1 , . . . , i r ∈ {1, 2} not all equal to 2, (29)

1

N 1≤n≤N µ i1 n+s1 • • • µ ir n+sr ----→ N →∞ 0.
The well-known Chowla conjecture asserts that, in the Möbius case, (µ n ) = (µ(n)) satisfies the above property.

Now we want to highlight the links between the Chowla property and the unpredictability of the sequence π := (π n ) n∈N * . Let us consider the most unpredictable shift-invariant probability measure on Y , namely the Bernoulli measure denoted by β under which the coordinates are independent and uniformly distributed on {-1, 1}. Assume that π is generic for β, i.e. that for each cylinder set C = {y ∈ Y :

y 1 = α 1 , . . . , y m = α m }, (30) 1 N 0≤n<N ½ C (S n π) ----→ N →∞ β(C) = 1 2 m .
Recall from Theorem 4.1 that the sequence η = µ 2 is generic for the shiftinvariant measure ν B . Then it follows that the pair (η, π) is generic for the product measure ν B ⊗ β on X × Y . Indeed, any weak limit ρ of a subsequence

  1 N j 0≤n<Nj δ (S n X η,S n Y π)   j≥1
has to be a joining of the two dynamical systems (X, ν B , S X ) and (Y, β, S Y ). But the former is a discrete-spectrum dynamical system, and the latter is a Bernoulli shift, hence these two systems are disjoint and ρ can only be the product measure ν B ⊗ β (see [START_REF] Furstenberg | Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation[END_REF]). As a consequence, since µ n = η n π n , we get that µ is generic for the probability measure ν M defined on Z as the pushforward measure of ν B ⊗ β by the continuous map Θ :

(x, y) ∈ X × Y → x • y ∈ Z where (x • y) n := x n y n .
We can easily compute the measure ν M of each cylinder set C = {z ∈ Z :

z 1 = α 1 , . . . , z m = α m } ⊂ Z.
First, let us denote by C 2 the associated cylinder set in X:

C 2 := {x ∈ X : x 1 = α 2 1 , . . . , x m = α 2 m }, and set λ(C) := 1≤n≤m α 2 n . Then we have ν M (C) = 1 2 λ(C) ν B (C 2 ).
Note that, in this situation, what we proved in the context of Ω is also valid in the context of Ω ′ .

Under the hypothesis (31), the Borel-Cantelli lemma ensures that ∆(ω ′ ) < ∞ P ′ -a.s., and we can consider the map ψ : Ω ′ → Y defined P ′ -almost surely by ψ(ω ′ ) := (-1) ∆(T ′ ω ′ ) , (-1) ∆(T ′2 ω ′ ) , . . . , (-1) ∆(T ′n ω ′ ) , . . . . Let us denote by ν ′ the pushforward measure of P ′ by ψ. Then ν ′ is a shiftinvariant probability measure on Y . Theorem 6.2. Assume that (31) holds. Then the sequence π is generic for ν ′ , i.e. for any cylinder set C, we have

(32) 1 N 0≤n<N ½ C (S n π) ----→ N →∞ ν ′ (C).
Proof. The arguments are similar to those in the proof of Theorem 4.1. Let B be a finite subset of N * , let (α m ) m∈B ∈ {-1, 1} B , and let C ⊂ {-1, 1} N * be the cylinder set

C := y = (y n ) n∈N * ∈ {-1, 1} N * : ∀m ∈ B, y m = α m .
We have to prove the convergence

1 N 0≤n<N ½ ψ -1 (C) (T ′n ω ′ ) ----→ N →∞ P ′ ψ -1 (C) . Set, for ω ′ ∈ Ω ′ and ℓ ≥ 1 ∆ ℓ (ω ′ ) := {k : 1 ≤ k ≤ ℓ, ω ′ k = 0}
. Under the assumption (31), ∆ ℓ (ω ′ ) ---→ ℓ→∞ ∆(ω) P ′ -a.s. Hence, considering the event

G ℓ := ω ′ ∈ Ω ′ : ∀m ∈ B, (-1) ∆ ℓ (T ′m ω ′ ) = α m , we have ½ G ℓ P ′ -a.s. ----→ ℓ→∞ ½ ψ -1 (C) ,
and by the dominated convergence theorem

P ′ (G ℓ ) ---→ ℓ→∞ P ′ ψ -1 (C) .
For the same reason as in [START_REF] Titchmarsh | The theory of the Riemann zeta-function[END_REF], we have

1 N 0≤n<N ½ G ℓ (T ′n ω ′ ) ----→ N →∞ P ′ (G ℓ ),
and it only remains to show that (33) sup Now, what can we say about the measure ν ′ ? Since the measurable dynamical system (Y, ν ′ , S) is a factor of (Ω ′ , P ′ , T ′ ), it has zero entropy, and we cannot hope that this measure ν ′ be the Bernoulli measure β. Moreover, observe that

N ≥1 1 N 0≤n<N ½ ψ -1 (C)△G ℓ (T ′n ω ′ ) ---→ ℓ→∞ 0. But, as in the proof of Theorem 4.1, if ω ′ ∈ ψ -1 (C) △ G ℓ , then there exist m ∈ B and k > ℓ such that ω ′ k + m = 0 [a k ]. We get, as in (15), ½ ψ -1 (C)△G ℓ ≤ m∈B k>ℓ f ′ m,k , where f ′ m,k (ω ′ ) := 1 if ω ′ k + m = 0 [a k ], 0
P ′ (∃k : ω ′ k = 0) ≤ k≥1 P ′ (ω ′ k = 0) = Σ.
Therefore, with probability at least 1 -Σ, there is no k such that ω ′ k = 0, hence (-1) ∆(ω ′ ) = 1 with probability at least 1 -Σ, which can be close to 1 if Σ is small. However, the following results show that, as we approach the Möbius case, this measure ν ′ for which π is generic looks more and more like the Bernoulli measure β. Proposition 6.3. As Σ → ∞, we have the convergence

P ′ (-1) ∆ = 1 -→ 1 2 . Proof. Since P ′ ω ′ k = 0 = 1/a k , denoting by E ′ [ • ]
the expectation with respect to the probability P ′ , we have

E ′ (-1) ½ ω ′ k =0 = 1 - 2 a k .
Then, by independence of the coordinates under P ′ , using the inequality ln(1 + x) ≤ x for -1 < x < 1, we get

E ′ (-1) ∆ = E ′ (-1) k≥1 ½ ω ′ k =0 = k≥1 E ′ (-1) ½ ω ′ k =0 = k≥1 1 - 2 a k ≤ exp(-2Σ).
The proposition follows, since the above expectation is also equal to

2P ′ (-1) ∆ = 1 -1.
Theorem 6.4. We have the weak convergence

ν ′ ----→ Σ→∞ β.
Proof. For m ≥ 1, let us call an m-cylinder any cylinder subset of Y of the form

C = {y ∈ Y : ∀1 ≤ j ≤ m, y j = α j }
for a fixed choice of (α 1 , . . . , α m ) ∈ {-1, 1} m . We have to prove that, for any m ≥ 1, and any m-cylinder C,

Σ→∞ ν ′ (C) = 1 2 m . (34) lim
We will prove this by induction on m, observing that Proposition 6.3 already gives the above convergence when m = 1. Assuming (34) holds for (m -1)-cylinders, we will get it for m-cylinders by showing that the ν ′ -distribution of y m conditioned on (y 1 = α 1 , . . . , y m-1 = α m-1 ) can be made arbitrarily close to the uniform distribution on {-1, 1} if Σ is large enough. But, for a random variable taking values in {-1, 1}, saying that its distribution is close to the uniform distribution on {-1, 1} amounts to saying that its expected value is close to 0. Hence, translated into the probability space (Ω ′ , P ′ ), what we want to prove is equivalent to

(35) lim Σ→∞ E ′ (-1) ∆•T ′m (-1) ∆•T ′ = α 1 , . . . , (-1) ∆•T ′m-1 = α m-1 = 0.
Recall that we assumed the validity of (34) for (m -1)-cylinders. Hence we can take Σ large enough so that

P ′ (-1) ∆•T ′ = α 1 , . . . , (-1) ∆•T ′m-1 = α m-1 ≥ 1 2 m . Then, (35) is equivalent to (36) E ′ (-1) ∆•T ′m ½ (-1) ∆•T ′ =α1,...,(-1) ∆•T ′m-1 =αm-1 ----→ Σ→∞ 0.
For j ∈ N * , we introduce the random subset K j = K j (ω ′ ) ⊂ {1, 2, 3, . . .}, defined as follows:

K j := k ≥ 1 : ω ′ k + j = 0 [a k ] , so that ∆ • T ′j = |K j |.
Conditioning with respect to (-1) ∆•T ′j , 1 ≤ j ≤ m -1 is not convenient: It is more appropriate to condition on a more precise information, namely the knowledge of K 1 , . . . , K m-1 .

Observe that the information provided by K 1 , . . . , K m-1 is precisely the following: For each k ≥ 1 and each 1 ≤ j ≤ m -1, we know whether ω ′ k + j = 0 [a k ] or not. Then, with respect to the conditional probability given K 1 , . . . , K m-1 , we have the three following properties:

• The coordinates ω ′ k , k ≥ 1 remain independent. • If k ∈ K j for some 1 ≤ j ≤ m -1, since we know that ω ′ k + j = 0 [a k ], then either j = m [a k ], and we have ω ′ k + m = 0 [a k ], or j = m [a k ], and we have ω ′ k + m = 0 [a k ]. In either case, we have

E ′ (-1) ½ ω ′ k +m=0 [a k ] K 1 , . . . , K m-1 = 1. • If k ∈ K 1 ∪ • • • ∪ K m-1
, then we know that ω ′ k + j = 0 [a k ] for each 1 ≤ j ≤ m -1. Hence a k > m -1, and ω ′ k is uniformly distributed on Z/a k Z \ {a k -1, . . . , a k -(m -1)}. We then have (37)

P ′ k ∈ K m |K 1 , . . . , K m-1 = 1 a k -m + 1
, and

E ′ (-1) ½ ω ′ k +m=0 [a k ] K 1 , . . . , K m-1 = 1 - 2 a k -m + 1
.

By the same argument as in the proof of Proposition 6.3, we get

E ′ (-1) ∆•T ′m K 1 , . . . , K m-1 = k≥1 k ∈K1∪•••∪Km-1 1 - 2 a k -m + 1 ≤ exp(-2F ), ( 38
)
where F is the random variable defined by

F := k≥1 k ∈K1∪•••∪Km-1 1 a k -m + 1
(the randomness of F coming from the role of the random sets K j ). Now, we have to prove that, if Σ is large, then F is large with probability P ′ close to 1.

Fix some small number ε > 0, and a number M large enough so that 2 M-1 < ε. We have

F ≤ k≥1 a k >m-1 1 a k -m + 1 = k≥1 m-1<a k ≤Mm 1 a k -m + 1 + k≥1 a k >Mm 1 a k -m + 1 .
The first sum can be bounded by 1≤ℓ≤Mm 1/ℓ, which is less than Σ/(M -1) if Σ is large enough. In the second sum, we have a k -m + 1 ≥ a k (M -1)/M , hence this second sum can be bounded by ΣM/(M -1). We then get, if Σ is large enough, (39)

F ≤ M + 1 M -1 Σ ≤ (1 + ε)Σ.
On the other side, by (37) we can interpret F as

F = E ′ |K m \ K 1 ∪ • • • ∪ K m-1 | K 1 , . . . , K m-1 .
Observe that, if k ∈ K m ∩ (K 1 ∪ • • • ∪ K m-1 ), then ω ′ k + m = 0 [a k ], and there exists also 1 ≤ j ≤ m -1 with ω ′ k + j = 0 [a k ]. It follows that j = m [a k ], and necessarily a k ≤ m. Since the integers a k are distinct, we deduce that

K m ∩ (K 1 ∪ • • • ∪ K m-1 ) ≤ m,
and it follows that

F ≥ E ′ |K m | K 1 , . . . , K m-1 -m.
Therefore, provided Σ is large enough, (40)

E ′ [F ] ≥ E ′ |K m | -m = Σ -m ≥ (1 -ε) Σ.
Then, using (39), we get

(1 -ε) Σ ≤ E ′ [F ] = E ′ [F ½ F ≤εΣ ] + E ′ [F ½ F >εΣ ] ≤ ε Σ + (1 + ε) Σ P ′ (F > εΣ),
and finally (41) P ′ (F > εΣ) ≥ 1 -2ε 1 + ε .

Now, let us come back to our goal, which is to establish (36). If Σ is large enough, we have E ′ (-1) ∆•T ′m ½ (-1) ∆•T ′ =α1,...,(-1) ∆•T ′m-1 =αm-1 = E ′ E ′ (-1) ∆•T ′m K 1 , . . . , K m-1 ½ (-1) ∆•T ′ =α1,...,(-1) ∆•T ′m-1 =αm-1

≤ E ′ E ′ (-1) ∆•T ′m K 1 , . . . , K m-1 ≤ E ′ [exp(-2F )] by (38)
≤ exp(-2εΣ) + P ′ (F ≤ εΣ).

By (41), the second term can be made as small as we want by taking ε small enough, then the first term can also be arbitrarily small by taking Σ large enough.

In view of Theorem 6.4, we are led to formulate the following conjecture which, by Theorem 6.1, would imply the validity of the Chowla conjecture. Conjecture 6.5. When Σ = ∞, the sequence π is generic for β.

Mariusz Lemańczyk: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, 12/18 Chopin street, 87-100 Toruń, Poland E-mail address: mlem@mat.umk.pl

1. 2 .

 2 B-free numbers. More generally than the square-free numbers, we consider all natural numbers which have no factors in a given subset B = {b k : k ≥ 1} ⊂ {2, 3, 4, . . .} satisfying (1) ∀1 ≤ k < k ′ , b k and b k ′ are relatively prime, and

2. 2 .

 2 Admissible sequences. Let A be a subset of N * . For each b ≥ 1, we denote by t(A, b) the number of classes modulo b in A: t(A, b) := z ∈ Z/bZ : ∃n ∈ A, n = z [b] .

For b ≥ 1

 1 and z ∈ Z/bZ, let us denote by π z,b the infinite arithmetic subsequence π z,b := {n ∈ N * : z + n = 0 [b]}. Fix k ≥ 1. By construction of ϕ, the sequence ϕ(ω) is equal to zero along the infinite arithmetic subsequence π ω k ,b k . The proof of Proposition 3.2 will consist in showing that, with probability one, there is no other arithmetic subsequence π z,b k with z ∈ Z/b k Z along which ϕ(ω) is equal to zero. It will follow that, with probability one, we can recover the coordinate ω k from ϕ(ω) by observing along which infinite arithmetic subsequence π z,b k the sequence ϕ(ω) vanishes. Lemma 3.3. For each k ≥ 1, the σ-algebra of T b k -invariant sets in (Ω, P, T ) coincides modulo P with the σ-algebra generated by the k-th coordinate ω k . Proof. Let us fix k ≥ 1, and consider for any

Corollary 4 . 2 .

 42 The subshift {S n η : n ∈ N * } generated by η coincides with the subshift X B of B-admissible sequences.

  B K -admissible words. On the other hand, any B K -admissible word can be obtained in this way, and since there are K k=1 b k different possible choices in Step 1, we get the other inequality.

  otherwise, and we can repeat word for word the end of the proof of Theorem 4.1.

By first choosing K large enough, and then letting n go to ∞, we get (28) lim sup

On the other hand, using the subadditivity of log 2 γ K :

and recalling from (25) that γ(n) = γ K(n) (n), we get, for all n ≥ 1,

Together with (28), this proves (27).

The computation of the topological entropy of X B has been done by Peckner in [START_REF] Peckner | Uniqueness of the measure of maximal entropy for the squarefree flow[END_REF] in the context of the square-free flow, that is when B is the set of squares of primes. In this paper, Peckner also proves that there exists a unique invariant probability measure on X B with maximal entropy, and describes the structure of the measurable dynamical system defined by this measure. Question 5.4. Can we generalize Peckner's results in the B-free numbers setting?

A generalized version of the Möbius function

In the case where B is the set of squares of prime numbers, we have η n = µ(n) 2 , where µ : n → µ(n) is the classical Möbius function:

) dn otherwise, where d n is the number of prime factors of n.

We would like to interpret also this Möbius function (or its counterpart in a more general context) in a similar dynamical setting.

In this purpose, we assume now that for each k ≥ 1, b k = a 2 k , where the integers a k , k ≥ 1, are pairwise relatively prime (one can have in mind that (a k ) is an increasing sequence of prime numbers for example).

We are interested in the sequence µ = (µ n ) n∈N * defined as follows: First, set, for each n ∈ N * ,

Next, define π n := (-1) δn and finally µ n := η n π n .

Theorem 6.1. We have (i) =⇒ (ii) ⇐⇒ (iii), where (i) The sequence π is generic for β.

(ii) The sequence µ is generic for ν M .

(iii) The sequence µ satisfies the Chowla property.

(Note that the equivalence between (ii) and (iii) has been stated in [START_REF] Sarnak | Three lectures on the Möbius function randomness and dynamics[END_REF] in the context of the classical Möbius function.)

Proof. We have already seen just before the theorem that (i) =⇒ (ii), and it only remains to prove the equivalence of (ii) and (iii).

Consider f : z ∈ Z → f (z) := z 1 , and notice that 1

If we assume that (ii) holds, then the above expression converges to the integral

But the first factor in the right-hand side is equal to 0 as soon as at least one of the exponents i j is equal to 1, hence we get (iii).

To obtain the other direction, note that (iii) tells us that we have a correct limit for the products f i1 • S s1 • . . . • f ir • S sr when the exponents are not all even. But in the case of even exponents, the limit is always correct by Theorem 4.1. Since the family of functions f i1 • S s1 • . . . • f ir • S sr (when r, the s j and the exponents i j vary) separates points and is closed under taking products, (ii) follows by using the Stone-Weierstrass theorem. 6.2. Genericity of the sequence π. Now, we consider the compact abelian group Ω ′ := k≥1 Z/a k Z, we denote by P ′ the Haar measure on Ω ′ , and by T ′ the addition of (1, 1, . . .) in Ω ′ . Observe that the sequence π = (π n ) n∈N * can also be observed in the dynamical system (Ω ′ , T ′ , P ′ ): Indeed, consider the map ∆ : Ω ′ → N * ∪ {∞} defined by ∆(ω ′ ) := {k ≥ 1 : ω ′ k = 0} . Then, denoting by ω ′ the point (0, 0, . . .) ∈ Ω ′ , we have for n ∈ N * δ n = ∆(T ′n ω ′ ), and π n = (-1) ∆(T ′n ω ′ ) .

The main difficulty lies in the fact that the quantity ∆(ω ′ ) can take infinite values: For example, ∆(ω ′ ) = ∞ (while ∆(T ′n ω ′ ) is finite for n ≥ 1). In particular, if k≥1 1/a k = ∞ (which happens in the Möbius case), by independence of the events (ω ′ k = 0), k ≥ 1, the second Borel-Cantelli lemma gives that ∆(ω ′ ) = ∞ for P ′ -almost all ω ′ . However, we can place ourselves in the case where (31) Σ := k≥1 1 a k < ∞.