Mehdi Rahim
email: rahim.mehdi@gmail.com

Philippe Ciuciu
email: philippe.ciuciu@cea.fr

Salma Bougacha
email: salmabougacha@hotmail.com

Impact of perceptual learning on resting-state fMRI connectivity: A supervised classification study

. The same individuals were then scanned in rs-fMRI. Here, in a supervised classification framework, we demonstrate that FC metrics computed on rs-fMRI data are able to predict the type of training the participants received. On top of that, we show that the prediction accuracies based on tangent embedding FC measure outperform those based on our recently developed multivariate wavelet-based Hurst exponent estimator [3], which captures low frequency fluctuations in ongoing brain activity too.

I. INTRODUCTION

Multisensory interactions are ubiquitous in cortex and sensory cortices may be supramodal ie, capable of functional selectivity irrespective of the sensory modality of inputs [START_REF] Pascual-Leone | The metamodal organization of the brain[END_REF], [START_REF] Renier | Cortical plasticity and preserved function in early blindness[END_REF]. In [START_REF] Zilber | Supramodal processing optimizes visual perceptual learning and plasticity[END_REF], we asked whether learning to discriminate visual coherence could benefit from supramodal processing. To this end, three groups of twelve participants were briefly trained to discriminate which of a red or green intermixed population of random-dot-kinematograms (RDKs) was most coherent in a visual display while being recorded with MEG (see Fig. 1). During training, participants heard no sound (V), congruent acoustic textures (AV) or auditory noise (AVn); importantly, congruent acoustic textures shared the temporal statisticsi.e. coherence -of visual RDKs. After training, the AV group significantly outperformed participants trained in V and AVn although they were not aware of their progress. In pre-and post-training blocks, all participants were tested without sound and with the same set of RDKs.

After the MEG experiment, the same individuals underwent a rs-fMRI scanning session of about 9 min to determine whether multi-perceptual learning impacted intrinsic brain activity and functional networks. These data allow us to investigate functional connectivity (FC) analysis. In contrast to [START_REF] Lewis | Learning sculpts the spontaneous activity of the resting human brain[END_REF], we could not contrast FC metrics before and after learning since no fMRI data were collected before. Instead, in this paper we focus on discriminating the three groups relying solely on FC metrics extracted after the learning process. Such a discrimination proves that multi-perceptual learning shapes ongoing brain activity in a different manner than purely visual one and hence confirms to some extent previous findings [START_REF] Lewis | Learning sculpts the spontaneous activity of the resting human brain[END_REF].

The paper is organized as follows. In Sec. II, we describe our learning MEG experiment [START_REF] Zilber | Supramodal processing optimizes visual perceptual learning and plasticity[END_REF]. In Sec. III, the proposed FC analysis pipeline is presented. Group-level results are reported in Sec. IV together with the prediction accuracies of supervised classifiers that perform binary classification (eg, AV-V) from FC measures as input features. Also, our results are compared in terms of classification performance with those recently derived on the same dataset from wavelet-based Hurst exponent estimates [START_REF] Pellé | Multivariate Hurst exponent estimation in fMRI. Application to brain decoding of perceptual learning[END_REF]. Conclusions are drawn in Section V. II. BRAIN NETWORKS DEFINITION FROM MEG DATA Fig. 2(top) provides an illustration of regions of interest (ROIs) in the sensor and source space showing significant post-training changes in neural responses after training in the V, AV and AVn groups. Significant changes in hMT+, V4, ITC and vlPFC were common to all groups whereas pSTS, mSTS, and AC were specific to the multisensory AV and AVn groups. As shown in Fig. 2(bottom), the network observed in post-multisensory training was more extensive than in purely visual training. Strikingly, the pattern of activation in both AVn and AV was notably reversed in several regions including pSTS, AC, mSTS and V4: this suggests selective modulations of these cortical regions based on the stimuli presented during training but also direct functional connectivity between these regions. No significant changes of activity were observed in these regions for the V group.

Common to all groups, hMT+ (supramodal coherence selectivity) and vlPFC (selective attention) showed discriminative cortical responses as a function of the learned coherence levels. Additionally, all groups showed an increased activity in ITC only for the easy coherence levels suggesting an improvement in color-motion binding. In hMT+, the increase spread of neural response was shared by V and AVn, whereas selective activity was seen for AV solely.

Furthermore, solely for AV, activity in multisensory cortices (mSTS, pSTS) correlated with post-training performances (see [START_REF] Zilber | Supramodal processing optimizes visual perceptual learning and plasticity[END_REF] for details); additionally, the latencies of these effects suggested feedback from vlPFC to hMT+ possibly First, they moved incoherently during a period ranging from 0.3 s to 0.6 s. Then, one population of dots starts moving coherently for 1 s. Next, the participant must press a button to decide whether the coherent dots population is red or green. The task difficulty was varied by changing the proportion of dots moving in the same direction (easy level: 95 %, very difficult level: 15 %). Bottom right: Acoustic stimuli delivered to individuals in the AV and AVn groups. In AV, an acoustic texture was designed to map the direction of coherent dots point-by-point. In AVn, only white noise was mapped point-wise to dots. mediated by temporal cortices in AV and AVn. Altogether, we interpret our results in the context of the Reverse Hierarchy Theory (RHT) of learning [START_REF] Ahissar | The reverse hierarchy theory of visual perceptual learning[END_REF] in which supramodal processing optimizes visual perceptual learning by capitalizing on sensory-invariant representations, here, global coherence levels across sensory modalities.

III. FUNCTIONAL CONNECTIVITY ANALYSIS

Whole brain fMRI BOLD data were collected on 36 individuals on a 3.0 Tesla Siemens Tim Trio scanner using a gradientecho EPI sequence. Scans were acquired at every TR = 1.05 s using the following parameters (TE = 30 ms, matrix size: 64×64, FOV=192×192 mm 2 , slice-thickness=6 mm). A single run of N = 514 scans of rs-fMRI was collected in each individual. The general pipeline we implemented for FC analysis of rs-fMRI data is depicted in Fig. 3. Hereafter, we summarize the most important aspects of each step.

A. Definition of Regions of interest

Regarding the ROI definition, we investigated two different strategies. The first set of ROIs was informed by MEG data and consisted of extracting 22 regions (11 in each hemisphere, see Fig. 2 for the 20 ROIs on top of which we added the vlPFC bilaterally) involved during perceptual learning in each individual. Once identified in each individual, we extracted the coordinates of these regions in the MNI space owing to spatial normalization of anatomical MRI (aMRI) data and coregistration of MEG data with the individual's aMRI. Next, spheres of 6-mm radius were defined around the centres given by the above-mentioned coordinates to extract fMRI signals.

The second strategy relied on more brain regions using the multi-subject dictionary learning (MSDL) atlas. It has been designed over the last years in our group [START_REF] Varoquaux | Multi-subject dictionary learning to segment an atlas of brain spontaneous activity[END_REF]. It has 39 ROIs grouped in several restingstate networks that cover the whole brain. It is available in https://team.inria.fr/parietal/research.

B. Preprocessing steps

As detailed in Fig. 3, fMRI time series were spatially preprocessed using a standard procedure: given the short TR value, slice timing correction was not applied. Instead, the scans were realigned for motion correction. Coregistration with anatomy was performed to ensure good match between anatomical and functional data. Spatial normalization was performed using the DARTEL toolbox. Last, spatial smoothing was applied to each volume using an isotropic Gaussian kernel with FWHM = 6 mm.

The presence of noise in fMRI signals and the absence of any experimental paradigm in resting-state recordings calls for an accurate control of putative bias which results in misleading connectivity between unrelated regions. To face this concern t Fig. 3. Proposed pipeline for analyzing rs-fMRI data from pre-processings (temporal and spatial including normalization using DARTEL in SPM12) to supervised classification (binary group comparisons like AV-V). The outcome of supervised classification allows us to unravel the discriminant functional interactions between groups. This pipeline was implemented in Python language using the nipype interface and the nilearn software (http://nilearn.github.io/). and improve results consistency, one needs to appropriately characterize and remove noise effects. Here, we defined a general linear model to regress out confounds accounting for head motion (3 translation and 3 rotation parameters per frame) and their derivatives as well as a linear detrending regressor to remove the global drift due to physiological and thermal noises. Finally, the time series within each ROI was band-pass filtered between 0.01 Hz and 0.1 Hz to reduce the effect of the low frequency drift and the high frequency noise.

C. Functional connectivity metrics

We investigated the use of four different connectivity metrics to probe functional interactions between ROIs from rs-fMRI data: Pearson correlation, sparse graph lasso, sparse group-covariance matrix and tangent embedding. Hereafter, we summarize these four estimators. We report the results we got using tangent embedding since this technique was the most accurate in terms of classification performance.

Consider regions i and j and denote by ρ the Pearson correlation coefficient, then ρ

= Ci,j √ Ci,i √ Cj,j
where covariance matrix C is known to be well approximated with the classical maximum likelihood estimator (MLE or "empirical covariance" ie 1 N X T X if X N ×p denotes the timeseries across p ROIs), provided the number of observations (here N , the number of scans) is large enough as compared to the number of features (here p, the number of voxels or ROIs). In this study when focusing on 22 ROIs only (p N ), the MLE can be a good estimator of the eigenvalues of the covariance matrix, and the precision matrix obtained from its inversion (K = C -1 ) may be accurate enough. However, when considering the MSDL atlas p and N are of the same order of magnitude, hence N < p(p + 1)/2 and the performance of the covariance MLE degrades. To avoid such inversion problem, one usually resorts to the regularization of the precision matrix. The latter is proportional to the partial correlation matrix which gives partial independence relationship. Hence, if two features are independent conditionally on the others, the corresponding coefficient in the precision matrix will be zero. This is why it makes sense to estimate a sparse precision matrix: by learning independence relations from the data, the estimation of the covariance matrix is better conditioned. As a consequence, we tested the Graph-lasso estimator that uses an 1 penalty to enforce sparsity on the precision matrix: the higher its α parameter, the sparser the precision matrix K is:

K = argmin K tr CK -log det K + α K 1 (1)
where K 1 is the sum of the absolute values of off-diagonal coefficients of K. The algorithm employed to solve this problem is the Graph-lasso algorithm proposed in [START_REF] Friedman | Sparse inverse covariance estimation with the graphical lasso[END_REF].

We also tested the group-sparse covariance estimator which consists of imposing the same structure of conditional independence across subjects ie, the zeros in the different precision matrices should be at the same positions. This was proposed in [START_REF] Varoquaux | Brain covariance selection: better individual functional connectivity models using population prior[END_REF] as a group-level extension of Graph-Lasso (1) by penalizing precision matrices using a mixed norm.

Another possible construction of group covariance matrix relies on the principle of maximum entropy, assuming that the individual covariance matrices (C s ) n s=1 derive from a generalized matrix-valued Gaussian distribution [START_REF] Pennec | A Riemannian framework for tensor computing[END_REF] with mean covariance matrix C * . The discrepancy of each C s from C * is measured by embedding the tangent space to the set of nonsingular covariances into symmetric matrices [START_REF] Varoquaux | Detection of brain functional-connectivity difference in post-stroke patients using group-level covariance modeling[END_REF]:

∀s = 1, • • • n : Log(C * -1 2 C s C * -1 2 ) ( 2
)
where Log denotes the matrix logarithm. Upper triangular coefficients of this discrepancy matrix are connectivity features.

D. Scale-free brain dynamics as alternative descriptors

In a previous work [START_REF] Pellé | Multivariate Hurst exponent estimation in fMRI. Application to brain decoding of perceptual learning[END_REF], we measured Hurst exponents on the same rs-fMRI dataset to assess the modulation of scale-free (1/f power spectrum) brain dynamics induced by perceptual learning. To this end, we used a univariate wavelet-based estimator to fit the linear slope on the wavelet-based power spectrum within the (0.01 -0.1) Hz range [START_REF] Veitch | A wavelet based joint estimator of the parameters of long-range dependence[END_REF]. Technically, we used Daubechies wavelets with 2 vanishing moments. We already demonstrated that these exponents are relevant features for classifying the type of training underwent by the participants.

E. Connectivity-based classification

We used the logistic regression classifier to predict binary outcomes and to compare each group pair such as (AV-V). Input features are functional connectivity matrices (lower triangular part of covariance matrix: p(p -1)/2 components per participant), yielding a feature matrix F = (f T 1 , ..., f T n ) T of dimension (n, p(p-1)/2). Logistic regression is a linear model where the weight vector w is estimated to fit the features F to the participants group y = (y 1 , ..., y n ) T ∈ {0, 1} n as:

w = argmin w∈R p(p-1)/2 , c∈R λ 2 w 2 2 + n s=1
log(e -ys(fsw+c) + 1),

where λ controls the regularization, c is the intercept, and ŵ are the estimated connectivity model coefficients to define the specific group of a given participant s. High values in w can be interpreted as discriminative connectivity components between groups.

In our experiments, functional connectivity metrics were computed on the whole dataset. Then, each group pair of 24 individuals (taken out of 36) was split in training and test sets. The training set was composed of subjects randomly chosen in both groups with the same occurrence proportions. The classifier was assessed by a cross-validation scheme: A stratified-shuffle split loop with 100 iterations and a test fold size of 25% of the whole dataset. Our implementation was based on the scikit-learn package.

IV. RESULTS

We applied the different sparse inverse covariance estimators to the 3 different groups of rs-fMRI data. In what follows, we report the best results we obtained with respect to our classification target, namely those derived from tangent embedding FC metrics (2) and the set of MEG-informed ROIs.

A. Within-group FC patterns

Statistical significant FC patterns are depicted in Fig. 4 for each group. Globally, one can notice that only positive interactions are retrieved in V whereas the converse finding is observed in AVn. A combination of positive and negative interactions is retrieved in AV. Importantly, left vlPFC is positively connected to V4 and hMT+ in V and AV, respectively. This confirms the particular role played by hMT+ in AV training. We did not report such involvement in other groups. Surprisingly, we observed significant negative interactions between AC and V1/V2 in AV in the left hemisphere whereas positive interaction is significantly recovered in V. This result may seem inconsistent with our previous observation in MEG (see Fig. 2) even though we have not performed FC analysis from MEG data so far. As first outlined in MEG, here from rs-fMRI data we also consistently observed the specific involvement of pSTS, mSTS and ITC through negative interactions with other regions in AV and AVn.

B. Between-group comparisons

Instead of reporting two-sample t-test for group comparison, we performed supervised classification to predict the type of training the individuals received from their FC features. In addition to the prediction accuracy, the logistic regression model permits to highlight between-group functional connectivity differences that are statistically significant.

Table I summarizes the results we got for AV-V comparison using different connectivity metrics and atlases (see Fig. 3). It illustrates that the optimal combination is achieved by using the selected ROIs as atlas and the tangent embedding as a connectivity metric. Fig. 5 shows the prediction accuracies 1for the 3 pairwise group comparisons, either by tangent embedding metric and wavelet-based Hurst exponent estimates computed over the same frequency range (0.01-0.1 Hz), see [START_REF] Pellé | Multivariate Hurst exponent estimation in fMRI. Application to brain decoding of perceptual learning[END_REF] for details. We observed similar trends: the AV training type is easier to discriminate from others. Interestingly, prediction accuracies based on FC features are improved by more than 15 % for every group comparison as compared to those relying on Hurst exponent estimates.

Beyond such improvements in terms of classification performance, statistically discriminative weights between V and AV in the logistic regression model are shown in Fig. 6. For ease of visualization, we provide both matrix and connectome views, the latter being a thresholded version of the former. When focusing on the connectome view, we observed that positive interactions are inter-hemispheric whereas negative ones are stick to the left hemisphere. Strikingly, the bilateral FEFs seem to bring the most salient FC differences between V and AV. Indeed, stronger interaction (in red) between hMT+ and FEF in AV might corroborate the selective increase of activity in hMT+ we observed on MEG data for this group. Also, larger covariation (in red) between FEF and AC in AV might reflect some binding mechanism between visual and auditory modalities that was trained in AV and not in V. The lower covariation (in blue) we found between V4 (colorform selectivity) and vlPFC (selective attention) in AV is compatible with the RHT since the individuals who performed purely visual training strengthened their connections between visual sensory inputs and decision-making areas. On the other hand, in the matrix view one can check that individuals in AV strengthened their connectivity betwen hMT+ (supramodal coherence selectivity) and vlPFC, which is a specificity of supramodal learning.

V. CONCLUSION

In this paper, we developed a functional connectivity analysis pipeline for rs-fMRI data that embeds different spatial/temporal preprocessings, different ROI definitions, complementary metrics for measuring interactions between regions and finally supervised classifiers. We assessed this pipeline on a MEG/fMRI dataset composed of 36 individuals who learned a visual discrimination task using either uni-or multi-perceptual training. We demonstrated that the prediction accuracy of auditory-visual training was reached with a rate of about 83 % as compared to purely visual training. This clearly outperforms our recent study [START_REF] Pellé | Multivariate Hurst exponent estimation in fMRI. Application to brain decoding of perceptual learning[END_REF] using Hurst exponents as input features to the classifier. Our results were achieved by optimally combining some specific pre-processings (DARTEL normalization, ...), the definition of MEG-informed ROIs and use of tangent-embedding FC metrics and logistic regression. They suggest the investigation of optimal combination of scale-free brain dynamics with pairwise interactions as proposed in the fractal connectivity model [START_REF] Ciuciu | Interplay between functional connectivity and scale-free dynamics in intrinsic fMRI networks[END_REF].

Fig. 1 .

 1 Fig. 1. Multi-perceptual learning experiment undertaken during MEG acquisition (90 min). Left: Pre-and post-training task blocks consisting of visual stimuli depicted on top-right. Three different groups of individuals performed either visual ( V), congruent auditory and visual (AV) or incongruent auditory and visual (AVn) training. Top right: Two random dot kinematograms (green and red) were presented to the individual.First, they moved incoherently during a period ranging from 0.3 s to 0.6 s. Then, one population of dots starts moving coherently for 1 s. Next, the participant must press a button to decide whether the coherent dots population is red or green. The task difficulty was varied by changing the proportion of dots moving in the same direction (easy level: 95 %, very difficult level: 15 %). Bottom right: Acoustic stimuli delivered to individuals in the AV and AVn groups. In AV, an acoustic texture was designed to map the direction of coherent dots point-by-point. In AVn, only white noise was mapped point-wise to dots.

Fig. 2 .

 2 Fig. 2. Top-left: Topography of ERF obtained in response to the presentation of incoherent visual RDKs is provided for the norm of gradiometers averaged over 100 to 300 ms post-incoherence onset (n = 36 subjects). Top-right: The corresponding current source estimates (MNE, dSPM) are provided. The extent of a given label or ROI in source space was defined by thresholding the dSPM estimates at the 90th percentile of all dSPM values. FEF: Frontal Eye Field. IPS: Inferior Parietal Sulcus. pSTS: posterior Superior Temporal Sulcus. AC: Auditory Cortex. mSTS: middle Superior Temporal Sulcus. ITC: Inferior Temporal Cortex. Bottom: ROIs where activity is specifically increased (red) or decreased (blue) in each group. Common changes are depicted in gray.

Fig. 4 .

 4 Fig. 4. Statistical significant functional interactions (positive and negative values are color coded in red and blue, respectively) within each group of individuals (V, AV and AVn from top to bottom), Bonferroni-corrected for multiple comparisons at α = 0.05.

Fig. 6 .

 6 Fig.6. Significant connectivity-based discriminative weights for the AV-V classification task. Positive/negative (red/blue) connections quantify betweengroup differences. Respectively, higher/lower connectivity in AV than V. Top: Matrix view with positive/negative weights. Bottom: Connectome view where only the most salient pairwise interactions are displayed. We observe that hMT+ is coded as MT. lh/rh stand for left and right hemispheres.

TABLE I MEANS

 I AND STANDARD DEVIATIONS OF PREDICTION ACCURACY (IN %)OF AV-V CLASSIFICATION ON TEST SETS ACROSS FOLDS.

			Tangent embed.	Correlation	Group sparse cov.
	MSDL	82.0 ± 15.9	63.0 ± 17.1	61.7 ± 19.8
	ROIs	86.0 ± 12.6	60.0 ± 18.3	72.3 ± 15.1
			Pairwise classification accuracies
		100
		80
	Accuracy (in %)	0 40 60 20	avv Hurst exponent Functional connectivity Pairwise groups avavn	vavn
	Fig. 5. Classification performances on functional connectivity mea-
	sures (based on tangent space estimator) and wavelet-based Hurst
	exponents as input features to discriminate pairwise groups (AV-V,
	AV-AVn and V-AVn). Median is depicted in dashed line, quartiles in
	dotted lines.

Box plots show the median and quartiles over 100 stratifications.

ACKNOWLEDGMENTS

The authors warmly thank Régine Bricquet, Alexandre Abraham, and our team leader Bertrand Thirion for organizing the 2015 Parietal retreat, during which this project started.


Karlheinz Gr
email: karlheinz.groechenig@univie.ac.at

Philippe Jaming
email: philippe.jaming@math.u-bordeaux.fr

THE CRAMER-WOLD THEOREM ON QUADRATIC SURFACES AND HEISENBERG UNIQUENESS PAIRS

Keywords: Mathematics Subject Classification. 42A68, 42C20. Key words and phrases. Uncertainty principle, annihilating pair, Heisenberg uniqueness pair, Cramér-Wold Theorem. 1

Two measurable sets S, Λ ⊆ R d form a Heisenberg uniqueness pair, if every bounded measure µ with support in S whose Fourier transform vanishes on Λ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in R d . As a corollary we obtain a new, surprising version of the classical Cramér-Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of a dense set of projections). We also give an application to the unique continuation of eigenfunctions of second-order PDEs with constant coefficients.

Introduction

The notion of a Heisenberg uniqueness pair by Hedenmalm and Montes-Rodrígez [HMR] introduced a new facet to the investigation of Fourier transform pairs (µ, µ) for a bounded Borel measure µ and its Fourier transform (or characteristic function)

µ(ξ) = R d e i x,ξ dµ(x), ξ ∈ R d .
In general a version of the uncertainty principle balances the size (of the support or of some quantity) of µ with the size of µ, so that both cannot be too small. By contrast, Heisenberg uniqueness pairs balance the size of the support of a measure µ with the size of the zero set of µ. A particular aspect is that both the support of µ and the zero set of µ may be singular sets.

1.1. Heisenberg Uniqueness Pairs.

To be precise, let S, Λ ⊆ R d be two measurable sets and let M(S) denote the set of finite signed Borel measures supported in S. The following definition was first introduced in [HMR] in a sightly more restrictive form.

Definition 1. The pair (S, Λ) ⊂ R d ×R d is said to be a Heisenberg uniqueness pair, if the only measure µ ∈ M(S) such that µ = 0 on Λ is the measure µ = 0.

For a subset C ⊂ M(S) of measures, (S, Λ) is said to be a C-Heisenberg uniqueness pair if the only measure µ ∈ C such that µ = 0 on Λ is the measure µ = 0.

Equivalently, if two finite measures supported on S have characteristic functions agreeing on Λ, then they are equal.

If S is a smooth manifold, we write AC(S) for the set of finite measures that are absolutely continuous with respect to the surface measure on S. The original concept in [HMR] was that of an AC(S)-Heisenberg uniqueness pair and so far was almost exclusively studied in dimension d = 2 when S is a smooth curve.

Since a general characterization of Heisenberg uniqueness pairs is out of reach, research so far has focussed on the investigation of specific examples, for instance S being a hyperbola [HMR], a circle [START_REF] Lev | Uniqueness theorems for Fourier transforms[END_REF][START_REF]Sjölin Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin[END_REF], a parabola [START_REF]Sjölin Heisenberg uniqueness pairs for the parabola[END_REF], and Λ a set of lines or a discrete subset thereof, some cases with three parallel lines are treated in [Ba], the case of both S and Λ being circles or spheres is treated in [START_REF] Lev | Uniqueness theorems for Fourier transforms[END_REF][START_REF]Sjölin Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin[END_REF][START_REF] Srivastava | Non-harmonic cones are Heisenberg uniqueness pairs for the Fourier transform on R d[END_REF].

In [JK] most of these examples were unified and extended by means of a new technique. The question of whether (S, ℓ 1 ∪ ℓ 2 ) is a AC(S)-Heisenberg uniqueness pair can then be answered by studying a certain dynamical system on S defined by two (distinct) lines Λ = ℓ 1 ∪ℓ 2 . * With this technique, one can derive a complete characterization in the case of conic sections S = {(x, y) ∈ R 2 : ax 2 + bxy + cy 2 + dx + ey = f } ⊆ R 2 with a, b, c, d, e, f ∈ R not all 0 and two lines Λ = ℓ 1 ∪ ℓ 2 . Before a description of these results, we make a convenient reduction. As already observed in [HMR], the notion of Heisenberg uniqueness pairs is invariant with respect to affine linear transformations, namely:

[Inv 1] Fix x, ξ ∈ R d . Then S, Λ is a Heisenberg uniqueness pair, if and only if Sx, Λξ is a Heisenberg uniqueness pair. [Inv 2] Fix T a linear invertible transformation R d → R d and denote by T * its adjoint. Then S, Λ is a Heisenberg uniqueness pair, if and only if T -1 (S), T * (Λ) is a Heisenberg uniqueness pair.

According to the invariance properties [Inv1-Inv2] it is enough to consider the following cases for the analysis of conic sections in R 2 .

(i) A single line. If Γ = {(x, y) ∈ R 2 : y = 0} and ℓ 1 , ℓ 2 are two arbitrary lines through (0, 0), then (Γ, ℓ 1 ∪ ℓ 2 ) is a Heisenberg uniqueness pair. Actually (Γ, ℓ 1 ) is already a Heisenberg uniqueness pair unless ℓ 1 is the line orthogonal to Γ [HMR]. (ii) The parabola. If Γ = {(x, y) ∈ R 2 : y = x 2 } and ℓ 1 , ℓ 2 are two arbitrary lines through (0, 0), then (Γ, ℓ 1 ∪ ℓ 2 ) is a Heisenberg uniqueness pair [START_REF] Lev | Uniqueness theorems for Fourier transforms[END_REF][START_REF]Sjölin Heisenberg uniqueness pairs for the parabola[END_REF].

(iii) The circle. If Γ = {(x, y) ∈ R 2 : x 2 + y 2 = 1} and ℓ 1 , ℓ 2 are two arbitrary lines through (0, 0) which intersect with an angle θ / ∈ πQ, then (Γ, ℓ 1 ∪ ℓ 2 ) is a Heisenberg uniqueness pair [START_REF] Lev | Uniqueness theorems for Fourier transforms[END_REF][START_REF]Sjölin Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin[END_REF]. † (iv) The hyperbola. If Γ = {(x, y) ∈ R 2 : xy = 1} and ℓ 1 , ℓ 2 are two lines through (0, 0). If

ℓ 1 = R(a, b) and ℓ 2 = R(-a, b), then (Γ, ℓ 1 ∪ ℓ 2 )
is a Heisenberg uniqueness pair [HMR]. Further degenerate quadratic curves which where not discussed in [JK], but for which the techniques of [JK] apply are the following.

(v) The cone

Γ = {(x, y) ∈ R 2 : xy = 0}. If ℓ 1 = R(a, b) and ℓ 2 = R(-a, b), then (Γ, ℓ 1 ∪ ℓ 2 ) is a Heisenberg uniqueness pair. (vi) Two parallel lines Γ = {(x, y) ∈ R 2 : x 2 = a 2 }. If ℓ 1 , ℓ 2 are
two arbitrary lines through (0, 0) except the y axis, (Γ, ℓ 1 ∪ ℓ 2 ) is a Heisenberg uniqueness pair. By applying the invariance properties of Heisenberg uniqueness pairs, we may summarize these results as follows.

Theorem 1.1. Let Q be a quadratic form on R 2 , v ∈ R 2 , ρ ∈ R, and S = {(x, y) ∈ R 2 : Q(x, y) + 2 v, (x, y) = ρ}. Then there exists an exceptional set E = E(Q, v, ρ) of pairs of distinct directions, such that when ℓ 1 , ℓ 2 are two lines with directions not in E, then (S, ℓ 1 ∪ℓ 2 ) is a Heisenberg uniqueness pair. Moreover, if ℓ 1 is fixed, then the set {ℓ 2 : (ℓ 1 , ℓ 2 ) /
∈ E} is at most countable. Therefore E has measure zero with respect to the surface measure on S 1 × S 1 .

Remark 1. The dependence of the exceptional set on the parameters Q, v, ρ is simple. For S a parabola, a point (= degenerate ellipse), or two parallel lines (= degenerate parabola), E is the empty set. In all other cases E(Q, v, ρ) is a fixed set of directions that depends only on Q, but not on v and ρ. Thus for a fixed quadratic form Q only a set E(Q) and ∅ may occur.

In this paper we present the first complete study of Heisenberg uniqueness pairs in higher dimensions and will extend Theorem 1.1 to higher dimensions for the case of a quadratic hypersurface S and a union of hyperplanes Λ. We then make the connection with the classical Cramér-Wold Theorem which characterizes probability measures by their projections onto hyperplanes. In contrast to general measures, a measure supported on a quadratic hypersurface is uniquely determined already by its projection to two generic hyperplanes. Finally, as in [HMR] we interpret the notion of Heisenberg uniqueness pair on a quadratic hypersurface as a statement about the solutions to a partial differential equation of order two with constant coefficients.

Let us now describe our results with more precision. To start, let S be a quadratic hypersurface, that is a surface of the form S = {x ∈ R d : P (x) = 0} where P is a polynomial of total degree 2. In the following we always write P as the sum of a quadratic form Q and an affine form. This means that there exists a bilinear form B :

R d × R d → R, such that Q(x) = B(x, x) and a vector v ∈ R d and ρ ∈ R, such that P (x) = Q(x) + 2 v, x -ρ.
The set Λ will consist of two distinct hyperplanes H 1 , H 2 . We describe a hyperplane by a normal vector u ∈ R d with unit norm |u| = 1 and the offset parameter s ∈ R, and define

H u,s = {x ∈ R d : x, u = s} and H u = H u,0 .
We first consider the case of two intersecting hyperplanes. Our main result is the following.

Theorem 1.2. Let Q be a quadratic form on R d , v ∈ R d , ρ ∈ R and S = {x ∈ R d : Q(x) + 2 v, x = ρ}. There exists an exceptional set E = E(Q, v, ρ)
of pairs of distinct directions such that (i) the set E has measure zero with respect to the surface measure on

S d-1 × S d-1 ; (ii) when u 1 , u 2 ∈ R d satisfy Q(u 1 ), Q(u 2 ) = 0 and (u 1 , u 2 ) / ∈ E, then (S, H u 1 ∪ H u 2 ) is a Heisenberg uniqueness pair.
This result is very general and the proof gives an explicit construction of the exceptional set in terms of the exceptional sets described in the special cases of Theorem 1.1. Also notice that Theorem 1.2 holds for arbitrary measures, whereas the cited results in dimension d = 2 were proved only for absolutely continuous measures. The proof of Theorem 1.2 will be given in Section 2. We will significantly extend the approach with dynamical systems [JK].

We will then focus on some specific cases and derive more precise results. We will also consider the case of parallel hyperplanes. In Section 3 we study the case when S is a cone or a hyperboloid in R d and Λ consists of two or more hyperplanes. In Section 4 we study the case when S is a sphere in R d and Λ consists either of two parallel hyperplanes or several hyperplanes with a common intersection. It is a curious feature of the dynamical systems approach that infinite Coxeter groups will occur naturally. Indeed, we will apply a deep theorem about such groups in the proof of Theorem 4.2.

Let us now see how our results can be interpreted in probability theory and PDEs.

A Cramér-Wold type theorem.

Let X, Y be two metric spaces and f : X → Y a Borel mapping. Recall that, if µ is a measure on X, the push-forward of µ by f is the measure f * µ on Y defined by f * µ(E) = µ f -1 (E) and that, for every continuous compactly supported function g on Y ,

(1.1) Y g(y) df * µ(y) = X g f (x) dµ(x).
The classical Cramér-Wold Theorem [CW] asserts that a probability measure on R d is uniquely determined by the set {π * µ : π a projection on a hyperplane through 0}, that is, if π * µ = π * ν for every π, or equivalently π * (µ-ν) = 0, then µ-ν = 0. This fact is easily proven as follows: if π is the projection on a hyperplane H, then the Fourier transform of π * (µν) in H is just the restriction of µν to H (see (ii) in Lemma 2.2). Therefore, π * (µν) = 0 for every H if and only if µν = 0 which occurs if and only if µν = 0. As µν is continuous one only needs a dense set of hyperplanes.

In general, not much more can be said, see e.g. [BMR, Gi, He, Re]. In particular, finitely many projections do never determine a measure on R d completely. However, if we restrict the support of a measure to a quadratic hypersurface, we obtain a completely different version of a Cramér-Wold theorem. The following statement is an immediate consequence of Theorem 1.2.

Theorem 1.3. Let Q be a quadratic form on R d , v ∈ R d , ρ ∈ R and S = {x ∈ R d : Q(x) + 2 v, x = ρ}. Then there exists an exceptional set E = E(Q, v, ρ) of measure zero in S d-1 × S d-1 with the following property: Let u 1 , u 2 ∈ R d be distinct directions, such that Q(u 1 ), Q(u 2 ) = 0 and (u 1 , u 2 ) / ∈ E. If µ, ν ∈ M(S) and π Hu 1 * µ = π Hu 1 * ν and π Hu 2 * µ = π Hu 2 * ν, then µ = ν.
Thus a finite measure supported on a quadratic hypersurface is determined uniquely by its projections to two generic hyperplanes.

Applications to linear PDEs.

Let P be a quadratic polynomial on R d and S = {x ∈ R d : P (x) = 0}. Replacing each variable x j by the partial derivative 1 i ∂ ∂x j , we obtain a second order partial differential operator P (D) with constant coefficients. If µ is a finite measure, then a basic formula about Fourier transforms states that

P (D) µ(ξ) = R d e i x,ξ P (x) dµ(x) .
Consequently, if supp µ ⊆ S, then

P (D) µ ≡ 0 ,
and thus µ is a (distributional) solution of this PDE, as was observed in [HMR]. Conversely, if P (D) µ ≡ 0, then supp µ ⊆ S. Let us give an example of the kind of results we can obtain:

Example 1. Let ∆ = d j=1 ∂ 2 ∂x 2 j
be the standard Laplacian on R d . The simplest bounded eigenfunctions of ∆ with eigenvalue ρ < 0 are given by e λ (x) = e i λ,x with |λ| 2 = -ρ. Now assume that |λ| 2 = |µ| 2 = -ρ and that e λ (x) = e µ (x) on some hyperplane H. Without loss of generality, we may assume that H = R d-1 ×{0} and then λ j = µ j for j = 1, . . . , d -1. The condition |λ| 2 = |µ| 2 then shows that there are still two possibilities, namely µ d = ±λ d . Uniqueness is guaranteed as soon as we assume that e λ (x) = e µ (x) on a second hyperplane H ′ that is not parallel to H.

With P (x) = -|x| 2ρ, the eigenvalue equation ∆u = ρu can be written as P (D)u = 0. Since e λ = δ λ and P (λ) = 0, δ λ is indeed supported on S = {x : P (x) = 0}. Theorem 1.2 then states that, for a generic pair of intersecting hyperplanes (H,

H ′ ), e λ (x) = e µ (x) on H ∪ H ′ implies λ = µ.
Of course, the result for the simple eigenfunctions e λ is valid for an arbitrary pair of intersecting hyperplanes. However, for more general eigenfunctions, this is no longer true and some restrictions apply to the hyperplanes for the result to be true.

The actual result is much more general: Theorem 1.2 directly yields the following property of solutions of homogeneous second order PDEs.

Theorem 1.4. Let Q be a quadratic form on R d , v ∈ R d , ρ ∈ R and P (x) = Q(x) + v, x -ρ. Let u 1 , u 2 ∈ S d-1 be such that Q(u 1 ), Q(u 2 ) = 0 and u 1 , u 2 ∈ E(Q, v, ρ).
Let µ be a finite measure and assume that u = µ solves the partial differential equation P (D)u = 0.

If u vanishes on the hyperplanes H u 1 and H u 2 , then u = 0.

In other words, a non-trivial eigenfunction of a second-order linear partial differential operator with constant coefficients cannot vanish on two generic hyperplanes. It remains to be seen whether a similar result holds for distributions instead of measures.

Quadratic Hypersurfaces and Hyperplanes

Measures on quadratic hypersurfaces with Fourier transform vanishing on a hyperplane.

In this section we build up the proof of Theorem 1.2 and first investigate measures on a quadratic hypersurface S whose Fourier transform vanishes on a single hyperplane. We start with some simple geometric observations. Lemma 2.1. Let B be a bilinear form on R d × R d and Q be the associated quadratic form Q(x) = B(x, x). Let u, v ∈ R d with |u| = 1 and let π u be the orthogonal projection on the hyperplane H u . Define the affine linear transformation

R Q u,v by (2.2) R Q u,v (x) := x -2 B(x, u) + v, u Q(u) u .
Fix x ∈ R d and consider the equation in y

(2.3) Q(y) + 2 y, v = Q(x) + 2 x, v π u (y) = π u (x) . -If Q(u) = 0, then (2.3) has two solutions, y = x and y = R Q u,v (x). -If Q(u) = 0 and B(x, u) + v, u = 0, then y = x is the only solution of (2.3).
Notation. If no confusion can occur, we will simply write R u,v = R Q u,v . Further, if v = 0 we will write R u = R u,0 . In this case, R u is a linear mapping and we define R * u to be its adjoint: R u x, y = x, R * u y for every x, y ∈ R d . Note that (R Q u,v ) 2 = Id and that for the case B(x, u) = x, u and v = 0, R is just the reflection of a vector x across the hyperplane H u . In general, R is a reflection across the hyperplane H u with respect to the quadratic form Q(x) as a "metric."

Proof. If π u (x) = π u (y) then y = x + tu for some t ∈ R, and thus

Q(y) = Q(x) + t 2 Q(u) + 2tB(x, u) and y, v = x, v + t u, v .
The identity Q(y) + 2 y, v = Q(x) + 2 x, v then reads as

t 2 Q(u) + 2tB(x, u) + 2t u, v = 0.
It remains to solve for t. If Q(u) = 0, there are two solutions. Either t = 0 and thus y = x,

or t = -2 B(x, u) + v, u Q(u) and thus y = R u,v (x). If Q(u) = 0
and B(x, u) + v, u = 0, then t = 0, and thus y = x is the only solution.

A measure supported on a quadratic hypersurface whose Fourier transform vanishes on a hyperplane must possess some symmetry property. The next lemma gives a precise formulation of these symmetries and is a crucial extension of Lemma 2.1 of [JK].

Lemma 2.2. Let B be a bilinear form on R d × R d and Q be the associated quadratic form Q(x) = B(x, x). Let u, v ∈ R d with |u| = 1, s > 0, ρ ∈ R, and let S = {x ∈ R d : Q(x) + 2 x, v = ρ}.
For µ ∈ M(S) let ν u,s be the measure dν u,s = e is x,u dµ. Then the following are equivalent:

(i) µ(ξ) = 0 for every ξ ∈ H u,s . (ii) π u * ν u,s = 0. Moreover, if Q(u) = 0, (i)-(ii) are equivalent to the following two properties: (iii) R u,v * ν u,s = -ν u,s . (iv) µ(ξ + su) = -e -2i u,v u,ξ Q(u) µ(R * u ξ + su) for every ξ ∈ R d . Proof. (i) ⇔ (ii) If ξ ∈ H u,s , then ξ = π u ξ + su, and thus µ(ξ) = R d e i x,ξ dµ(x) = R d e i πux,πuξ e is x,u dµ(x).
Therefore, if we denote by ν the measure defined by dν(x) = dν u,s (x) = e is x,u dµ(x), then, according to (1.1),

µ(ξ) = R d e i πux,πuξ dν(x) = Hu e i y,πuξ dπ u * ν(y) = π u * ν(π u ξ)
where π u * ν is the characteristic function of the measure π u * ν defined on H u (identified with R d-1 ).

Since π u is a bijection from H u,s to H u , µ = 0 on H u,s if and only if π u * ν = 0 on H u that is π u * ν = 0. This establishes the equivalence of (i) and (ii).

(ii

) ⇒ (iii) Let E u = {x ∈ S : B(x, u) = -v, u }. If x ∈ E u , then R u,v (x) = x, consequently π u is one-to-one on E u and ν| Eu = R u,v * ν| Eu .
On the complement S \ E u the projection π u is two-to-one. We can therefore partition S \ E u = S + ∪ S -into two Borel sets S + , S -in such a way that π u is one-to-one on S + and on S -. Now assume that π u * ν = 0. Then ν| Eu = 0, which we may write as

ν| Eu = -R u,v * ν| Eu . Then 0 = Hu g(y) dπ u * ν = S g π u (x) dν(x) = Eu g π u (x) dν(x) + S + g π u (x) dν(x) + S - g π u (x) dν(x) = S + g π u (x) dν(x) + S - g π u (x) dν(x). (2.4) Since Q(u) = 0, Lemma 2.1 asserts that if x ∈ S ± then R u,v (x) ∈ S ∓ and π u (x) = π u R u,v (x) , and R 2 u,v (x) := R u,v R u,v (x) = x. But then S - g π u (x) dν(x) = S - g π u R 2 u,v (x) dν(x) = S + g π u R u,v (x) dR u,v * ν(x) = S + g π u (x) dR u,v * ν(x).
Therefore (2.4) reads as

(2.5)

S + g π u (x) d[ν(x) + R u,v * ν(x)] = 0.
Since every (continuous) function on S + can be written in the form g π u (x) for some (continuous) function g on H u , we get ν u,s = -R u,v * ν u,s on S + . Replacing S + by S -in the argument shows that ν u,s = -R u,v * ν u,s on S -as well.

(iii) ⇒ (ii) Conversely, assume that ν u,s + R u,v * ν u,s = 0. Since ν Eu = R u,v * ν| Eu by symmetry, it follows that ν u,s | Eu = 0, consequently π u * ν| Eu = 0. Reading (2.5) and (2.4) backwards, we find that also S\Eu g(π u (x))dν(x) = 0 and thus π u * ν = 0.

(iii) ⇔ (iv) Finally we note that

µ(ξ + su) = S e i x,ξ+su dµ(x) = S e i x,ξ dν(x) = ν(ξ),
whereas

S e i x,ξ dR u,v * ν(x) = S e i Ru,vx,ξ dν(x) = e -2i u,v u,ξ Q(u) S e i Rux,ξ dν(x) = e -2i u,v u,ξ Q(u) S e i x,R * u ξ+su dµ(x) = e -2i u,v u,ξ Q(u) µ(R * u ξ + su).
It follows that (iii) and (iv) are equivalent.

Corollary 2.3. Let B be a bilinear form on R d × R d and Q be the associated quadratic form

Q(x) = B(x, x). Let u, v ∈ R d with |u| = 1, s > 0, ρ ∈ R, and let S = {x ∈ R d : Q(x) + 2 x, v = ρ}. (i) (S, H u,s
) is a Heisenberg uniqueness pair if and only if π u is one-to-one on S.

(ii) If Q(u) = 0, (S, H u,s ) is not a Heisenberg uniqueness pair. (iii) Assume that Q(u) = 0 and µ ∈ M(S). Then µ vanishes on H u,s , if
and only if µ is supported on the set

E u = {x ∈ S : B(x, u) = -v, u }
and Eu e is x,u dµ(x) = 0. In particular, if E u ∩ S has measure zero with respect to the surface measure on S, then (S, H u,s ) is an AC(S)-Heisenberg uniqueness pair.

Proof. (i) Assume π u is one-to-one on S and µ ∈ M(S). Lemma 2.2 shows that if µ = 0 on H u,s then π u * ν u,s = 0 with dν u,s (x) = e is x,u dµ(x). Since π u is one-to-one, it follows that ν u,s = 0, thus µ = 0 and (S, H u,s ) is a Heisenberg uniqueness pair. Conversely, if π u is not one-to-one on S, there exist points x, y ∈ S, x = y with π u (x) = π u (y). Let µ = e -is x,u δ x -e -is y,u δ y = 0, so that ν u,s = δ x -δ y . Therefore π u * ν u,s = 0 and Lemma 2.2 shows that µ = 0 on H u,s .

(ii) If Q(u) = 0 and x ∈ S satisfies B(x, u) + v, u = 0, then Lemma 2.1 shows that y = R u,v x ∈ S \ {x} and π u (x) = π u (y). The previous argument shows that (S, H u,s ) is not a Heisenberg uniqueness pair. The proof easily adapts to show that (S, H u,s ) is not an AC(S)-Heisenberg uniqueness pair.

(iii) If Q(u) = 0, then π u : S → H u is one-to-one on S \ E u by Lemma 2.1. Therefore if π u * ν u,s = 0 then ν u,s = 0 on S \ E u and thus µ = 0 on S \ E u .

As a single hyperplane leads to a Heisenberg uniqueness pair only in exceptional cases, we are led to ask when (S,

H u 1 ,s 1 ∪ H u 2 ,s 2 ∪ • • • ∪ H u N ,s N ) for N ≥ 2
can be a Heisenberg uniqueness pair. In this case, the measure µ must satisfy multiple symmetries, and those symmetries may be incompatible. Our aim is to show that this is indeed the case. To do so, we will first investigate measures on S that have Fourier transform vanishing on two intersecting hyperplanes.

2.2. Proof of Theorem 1.2. Measures on quadratic surfaces with characteristic function vanishing on two hyperplanes.

We now prove Theorem 1.2. Throughout this section, Q, B and S will be as above. We will prove that, unless (u 1 , u 2 ) is in an exceptional set E of measure 0, a measure µ ∈ M(S) such that µ = 0 on

H u 1 ∪ H u 2 is necessarily µ = 0.
We start with an appropriate parametrization of S d-1 × S d-1 . For given u 1 ∈ S d-1 , every u 2 ∈ S d-1 can be written as

u 2 = cos θ u 1 + sin θ v 2 with unique v 2 ∈ S d-1 ∩ {u 1 } ⊥ (a d -2 dimensional sphere) and θ ∈ [0, π]. Note that span(u 1 , u 2 ) = span(u 1 , v 2 ).
Then the Lebesgue measure on S d-1 × S d-1 decomposes as

S d-1 ×S d-1 ϕ(u 1 , u 2 ) dσ d (u 1 ) dσ d (u 2 ) = S d-1 S d-1 ∩{u 1 } ⊥ π 0 ϕ(u 1 , cos θ u 1 +sin θ v 2 ) sin d-2 θ dθ dσ d-1 (v 2 ) dσ d (u 1 ) .
Indeed, this is just a way to write dσ d (u 2 ) in spherical coordinates.

To see that the exceptional set E has measure zero, we will show that, for u 1 , v 2 fixed, the set of angles θ for which (u 1 , cos θu 1 + sin θv 2 ) ∈ E is countable.

Note also that S Q := {u ∈ S d-1 : Q(u) = 0} is a lower dimensional submanifold of S d-1 and has therefore measure 0 in S d-1 , and so does

S Q × S Q in S d-1 × S d-1 .
From now on, we fix

u 1 ∈ S d-1 , v 2 ∈ S d-1 ∩ {u 1 } ⊥ and write u 2 = u 2 (θ) = cos θ u 1 + sin θ v 2 . We further assume that u 1 , u 2 / ∈ S Q and u 1 = u 2 .
Our aim is to show that there is an at most countable set of θ's for which (S, H u 1 ∪H u 2 ) is not an Heisenberg uniqueness pair.

Step 1. Let µ ∈ M(S) be such that µ = 0 on H u 1 ∪ H u 2 . We will write R j = R Q u j for the involutions on S defined by u 1 and u 2 . Then the equivalence (i) ⇔ (iii) of Lemma 2.2 implies that

R j * µ = -µ, j = 1, 2 thus (R 2 • R 1 ) * µ = µ.
Let π be the orthogonal projection on the subspace span(u 1 , u 2 ) generated by u 1 , u 2 and π ⊥ = Iπ be the orthogonal projection on the orthogonal complement of span(u 1 , u 2 ). Let S 0 be a measurable subset of S such that π ⊥ is one-to-one from S 0 to span(u 1 , u 2 ) ⊥ . Consequently, we may write every y ∈ S as y = x + u for a unique x ∈ S 0 and u ∈ span (u 1 , u 2 ).

We now make to crucial observations: (i) The intersection of S with the affine plane x+span (u 1 , u 2 ) is a conic section (an ellipse, parabola, hyperbola, or two lines), and (ii) every such cross section S∩(x+span (u 1 , u 2 )) is invariant under the involutions R 1 and R 2 . This follows from Lemma 2.1.

We may therefore use the analysis of Theorem 1.1 for each section and each pair of lines x + Ru 1 and x + Ru 2 . To apply this analysis, we further need an appropriate restriction of the given measure µ to the cross section S ∩ (x + span (u 1 , u 2 )). Technically, this is done by desintegration.

Let ν = π ⊥ * µ and let us write the disintegration theorem (e.g., [DM]) in the form

(2.6) S ϕ(x) dµ(x) = S 0 (π ⊥ ) -1 (x) ϕ(y) dµ x (y) dν(x) ϕ ∈ C c (S).
Recall that the measure µ x is uniquely determined by this formula for νalmost every x and that µ x is supported on S ∩ x + span(u 1 , u 2 ) .

Since π ⊥ u j = 0 for j = 1, 2, (2.2) implies that π ⊥ R j = π ⊥ . Therefore, for ϕ ∈ C c (S), we have

S ϕ(x) dR j * µ(x) = S ϕ R j (x) dµ(x) = S 0 (π ⊥ ) -1 (x) ϕ R j (y) dµ x (y) dν(x) = S 0 (π ⊥ ) -1 (x) ϕ(y) dR j * µ x (y) dν(x). (2.7) It follows that R j * µ = -µ if and only if R j * µ x = -µ x for ν-almost every x. The equivalence (i) ⇔ (iii) of Lemma 2.2 implies that µ x = 0 on Ru 1 ∪ Ru 2 . ‡
We may summarize this reduction to the two-dimensional case as follows.

Lemma 2.4. Let µ ∈ M(S) and u 1 , u 2 ∈ S d-1 . Then µ vanishes on H u 1 ∪ H u 2 , if and only if µ x vanishes on Ru 1 ∪ Ru 2 for ν-almost all x ∈ S 0 .

Next, for x ∈ S 0 the measure µ x is supported on the intersection of the quadratic surface S with the plane x + span(u 1 , u 2 ) = x + span(u 1 , v 2 ). This intersection is either the union of two lines, a parabola, a hyperbola or an ellipse. For each of these, we know from Theorem 1.1 when the condition µ x | Ru 1 ∪Ru 2 = 0 implies that µ x = 0.

Step 2. We now apply Theorem 1.1 to S ∩ (x + span(u 1 , u 2 )) and µ x . We first determine the intersection S x = S ∩ x + span(u 1 , u 2 ) = S ∩ x + span(u 1 , v 2 ) precisely.

As x ∈ S, a point x + su 1 + tv 2 belongs to S, if and only if

Q(x + su 1 + tv 2 ) + 2 x + su 1 + tv 2 , v = Q(x) + 2 x, v = ρ ,
or equivalently,

(2.8)

s 2 Q(u 1 ) + 2stB(u 1 , v 2 ) + t 2 Q(v 2 ) + 2sb(x) + 2tb ′ (x) = 0. with b(x) = B(x, u 1 ) + u 1 , v and b ′ (x) = B(x, v 2 ) + v 2 , v .
Let Σ be the set of points (s, t) ∈ R 2 satisfying (2.8). Then Σ is either a hyperbola, an ellipse, or a parabola in R 2 (where a degenerate hyperbola is a set of two intersecting lines and a degenerate ellipse is a point or the empty set). The classification into these three types depends on the value of the discriminant of the quadratic form Q

(s, t) = s 2 Q(u 1 )+2stB(u 1 , v 2 )+t 2 Q(v 2 ), namely ∆ = det Q(u 1 ) B(u 1 ,v 2 ) B(u 1 ,v 2 ) Q(v 2 ) = Q(u 1 )Q(v 2 ) -B(u 1 , v 2 ) 2 . The set S x = S ∩ x + span(u 1 , v 2 ) = {x + su 1 + tv 2 : (s, t) ∈ Σ} ⊆ R d
is obtained from Σ by a linear transformation and a shift by x and thus represents the same conic section as Σ. Specifically, if Q(u 1 )Q(v 2 )-B(u 1 , v 2 ) 2 < 0, ‡ The Fourier transform of µ x is here taken in x + span(u 1 , u 2 ). That is

µ x (ξu 1 + ηu 2 ) = R 2 e i(sξ+tη) dµ x (x + su 1 + tu 2 ).
then, for all x ∈ S 0 , S ∩ x + span(u 1 , v 2 ) is a hyperbola or two intersecting lines (in the degenerate case).

If Q(u 1 )Q(v 2 ) -B(u 1 , v 2 ) 2 > 0, then S ∩ x + span(u 1 , v 2 ) is an ellipse or a point (in the degenerate case). If Q(u 1 )Q(v 2 ) -B(u 1 , v 2 ) 2 = 0, then S ∩ x + span(u 1 , v 2
) is a parabola or a set of two parallel lines or a single line (in the degenerate case).

Theorem 1.1 states the existence of a set

E x = E( Q) ⊆ span(u 1 , v 2 ) of pairs of directions such that if (w 1 , w 2 ) / ∈ E x then µ x = 0. At this point it is crucial that E x depends only on Q, but not on x ∈ S 0 .
Consequently, if w 1 , w 2 ∈ E( Q), then µ x = 0 for all x ∈ S 0 . By (2.6) this implies that µ = 0.

Step 3. We finally calculate the measure of

E in S d-1 × S d-1 . Fix u 1 and v 2 ∈ u ⊥
1 , and take w 1 = u 1 and w 2 = cos θ u 1 + sin θ v 2 . Then the set of θ, such that w 1 , w 2 ∈ E x is contained in a fixed countable set independent of x. Consequently the measure of the exceptional set E is (2.9)

S d-1 ×S d-1 χ E (u 1 , u 2 ) dσ d (u 1 ) dσ d (u 2 ) = S d-1 S d-1 ∩{u 1 } ⊥ π 0 χ E (u 1 , cos θ u 1 +sin θ v 2 ) sin d-2 θ dθ dσ d-1 (v 2 ) dσ d (u 1 ) = 0 .
This completes the proof ot Theorem 1.2.

The cone and the hyperboloid

In this section we investigate special arrangements of hyperplanes for S being a hyperboloid or a cone.

Let p be an integer, 1 ≤ p ≤ d -1 and q = dp. Let Q be the quadratic form

Q(x) = x 2 1 + • • • + x 2 p -x 2 p+1 -• • • -x 2 p+q
with the associated bilinear form B and set v = 0. Let ρ ∈ R and

S ρ = {x ∈ R d : Q(x) = ρ}.
Then S 0 is a cone and S ρ , ρ = 0, is a hyperboloid with one or two connected components. For instance, in dimension 3, if p = 1 and q = 2 then S 1 has 2 sheets while S -1 has only one.

Our aim is to complement the statement of Theorem 1.2 which treats the case of pairs of intersecting hyperplanes H u 1 ∪H u 2 with non-isotropic normals, Q(u 1 ), Q(u 2 ) = 0. We will treat two cases. On one hand, we will consider the case of parallel hyperplanes and show that, generically, (S, H u,s 1 ∪H u,s 2 ∪H u,s 3 ) is a Heisenberg uniqueness pair. On the other hand, we will also show that it is possible to construct Heisenberg uniqueness pairs of the form (S,

H u 1 ∪• • • H u k )
where all normals u 1 , . . . , u k are isotropic vectors.

In this section, we will write R d = R p × R q in the obvious sense: if x ∈ R d , then x ′ is its projection on R p and x ′′ its projection on R q so that x = (x ′ , x ′′ ). In particular, x, u = x ′ , u ′ + x ′′ , u ′′ and B(x, u) = x ′ , u ′x ′′ , u ′′ . and on the other hand

µ(ξ + s 0 u) = µ(ξ + (s 0 -s)u + su) = -µ R * u (ξ + (s 0 -s)u) + su = -µ ξ + (s 0 -s)π u ⊥ ũ + (2s -s 0 )u . (3.12)
Now assume that µ = 0 on H u,s 0 . Then (3.11) implies that µ = 0 on H u,-s 0 . Applying now (3.12) with -s 0 instead of s 0 , we obtain that µ = 0 on H u,s 0 +2s . By induction, it follows that µ = 0 on H u,s 0 +2ks , for every k ∈ N. Reversing the order in which we use (3.11)-(3.12) we obtain that µ = 0 on H u,s 0 +2ks for every k ∈ Z.

Next, using the third symmetry we obtain that µ = 0 on H u,s 0 +2ks+2ℓt for every k, ℓ ∈ Z. As µ = 0 on H u,0 we get that µ = 0 on every hyperplane H u,2ks+2ℓt for every k, ℓ ∈ Z. This set is dense in R d and µ is continuous, therefore µ = 0 and (S, H u,0 ∪H u,s ∪H u,t ) is a Heisenberg uniqueness pair.

Intersecting hyperplanes with isotropic normal.

Again, let 1

≤ p ≤ q ≤ d -1, p + q = d with p + q = d, Q(x) = x 2 1 + • • • + x 2 p -x 2 p+1 -• • • -x 2 p+q , and S ρ = {x ∈ R d : Q(x) = ρ} for ρ ∈ R. We now assume that u ∈ R d is isotropic, i.e., Q(u) = 0. Write u = (u ′ , u ′′ ) ∈ R p × R q , then Q(u) = 0 is equivalent to |u ′ | = |u ′′ |. Without loss of generality, we can restrict our attention to u with |u ′ | = |u ′′ | = 1. Let µ ∈ M(S ρ ). By Corollary 2.3, μ vanishes on H u , if and only if µ is supported on the set E u = {x ∈ S ρ : B(x, u) = 0}. We represent E u in a different form as follows: a point (x ′ , x ′′ ) is in S ρ , if and only if |x ′ | 2 -|x ′′ | 2 = ρ, i.e., x ′ ∈ (|x ′′ | 2 + ρ) 1/2 S p-1 , and B(x, u) = 0, if and only if x ′ , u ′ = x ′′ , u ′′ . Consequently, (3.13) E u = (x ′ , x ′′ ) ∈ R p+q : x ′ ∈ (|x ′′ | 2 + ρ) 1/2 S p-1 ∩ x ′′ , u ′′ u ′ + (u ′ ) ⊥ .
We note right away that E u is symmetric, and thus x ∈ E u implies that -x ∈ E u .

This leads to the first obversation on intersecting hyperplanes with isotropic normals.

Proposition 3.2. (i) If Q(u) = 0 for non-zero u, then (S ρ , H u ) is an AC(S)-Heisenberg uniqueness pair. (ii) Given vectors u j , j = 1, . . . , k, such that Q(u 1 ) = • • • = Q(u k ) = 0, (S ρ , H u 1 ∪ • • • ∪ H u k ) is a Heisenberg uniqueness pair, if and only if k j=1 E u j is either empty or {0}. (iii) If k < p and Q(u 1 ) = • • • = Q(u k ) = 0, then (S ρ , H u 1 ∪ • • • ∪ H u k ) is not a Heisenberg uniqueness pair. Proof. (i) Since E u is a (d -2)-dimensional submanifold of the (d -1)- dimensional manifold S ρ
, it has surface measure zero in S ρ . An absolutely continuous measure with support in E u is necessarily 0 (Corollary 2.3(iii)).

(ii) Next assume that u 1 , . . . , u k are such that

Q(u 1 ) = • • • = Q(u k ) = 0. By Corollary 2.3, μ vanishes on k j=1 H u j , if and only if µ is supported on k j=1 E u j .
If this intersection is empty, then obviously µ = 0. If this intersection is {0}, then µ = cδ 0 . As µ(0) = 0, we obtain µ = 0. Note that if ρ = 0, then 0 / ∈ S ρ and we obtain directly that µ = 0. If k j=1 E u j contains at least two points, then there is a non-zero x such that {x, -x} ⊂ k j=1 E u j by the symmetry of E u . Now set µ = δ xδ -x ∈ M(S). Then µ is non-zero, but µ = 0 on

H u 1 ∪ • • • ∪ H u k , so (S ρ , H u 1 ∪ • • • ∪ H u k ) is not a Heisenberg uniqueness pair.
(iii) If k < p, then there exists a non-zero

x ′′ ∈ (u ′′ 1 ) ⊥ ∩ • • • (u ′′ k ) ⊥ of norm |x ′′ | 2 > |ρ| and a x ′ ∈ (|x ′′ | 2 + ρ) 1/2 S p-1 ∩ k j=1 (u ′ j ) ⊥ . Consequently E u 1 ∩ • • • ∩ E u k is not empty and contains the non-zero point (x ′ , x ′′ ). By item (ii) (S ρ , H u 1 ∪ • • • ∪ H u k ) is not a Heisenberg uniqueness pair.
Using the geometric characterization (ii) of Proposition 3.2, one can now derive numerous examples of Heisenberg uniqueness pairs. As an example we prove the following statements.

Proposition 3.3. Let ρ ∈ R, 1 ≤ p ≤ q ≤ d -1 with p + q = d, Q be a quadratic form Q(x) = x 2 1 + • • • + x 2 p -x 2 p+1 -• • • -x 2
p+q and S ρ be the cone (ρ = 0) or hyperboloid (ρ = 0)

S ρ = {x ∈ R d : Q(x) = ρ}. (i) If ρ > 0, there exist u 1 , . . . , u p = 0 with Q(u 1 ) = • • • = Q(u p ) = 0, such that (S 0 , H u 1 ∪ • • • ∪ H up ) is a Heisenberg uniqueness pair. (ii) If ρ = 0, there exist 2p vectors u 1 , . . . , u 2p = 0 with Q(u 1 ) = • • • = Q(u 2p ) = 0 such that (S 0 , H u 1 ∪ • • • ∪ H u 2p
) is a Heisenberg uniqueness pair. (iii) If ρ < 0, there exist p + q vectors u 1 , . . . , u p+q = 0 with Q(u 1 ) =

• • • = Q(u p+q ) = 0 such that (S 0 , H u 1 ∪ • • • ∪ H u p+q ) is a Heisenberg uniqueness pair.
Proof. Choose an orthonormal basis u ′ 1 , . . . , u ′ p of R p . Define u ′′ j = (u ′ j , 0, . . . , 0) ∈ R q , u j = (u ′ j , u ′′ j ), and ũj = (u ′ j , -u ′′ j ), j = 1, . . . , p. First note that, for every a 1 , . . . , a p , the intersection

a 1 u ′ 1 + (u ′ 1 ) ⊥ ∩ • • • ∩ a p u ′ p + (u ′ p ) ⊥ contains exactly one point, namely, x ′ = p j=1 a j u ′ j . If (x ′ , x ′′ ) ∈ ∩ j=1,...,p E u j then, in view of (3.13), x ′ ∈ ∩ p j=1 x ′′ , u ′′ j u ′ j + (u ′ j ) ⊥ , so that x ′ = p j=1 x ′′ , u ′′ j u ′ j and |x ′ | 2 = p j=1 x ′′ , u ′′ j 2 .
In particular, if we write x ′′ = (y, ỹ) with y ∈ R p and ỹ ∈ R q-p (with the obvious abuse of notation when q = p), then x ′ = y.

Let us first assume that ρ > 0 then

|x ′ | ≤ |x ′′ | < (|x ′′ | 2 + ρ) 1/2
. This contradicts (3.13) and thus implies that j=1,...,p E u j = ∅. Applying Proposition 3.2, this proves (i).

If ρ ≤ 0, j=1,...,p

E u j = {(x ′ , x ′ , x), x ′ ∈ R p , x ∈ R q-p } ,
and similarly, j=1,...,p

E ũj = {(x ′ , -x ′ , x), x ′ ∈ R p , x ∈ R q-p } .
If q = p, then already j=1,...,p E u j ∩ j=1,...,p E ũj = {0}.

For ρ = 0 the condition |x ′ | = |x ′′ | = |(x ′ , x)| implies that x = 0,
E ũj = {(0, . . . , 0, x), x ∈ R q-p }.
Complete the orthonormal set u ′′ j , j = 1, . . . , p with vectors u ′′ j , j = p + 1, . . . , q, into an orthonormal basis of R q and let u j = (u ′ 1 , u ′′ j ), j = p+1, . . . , q. If x ∈ ∩ j=1,...,q E u j ∩ ∩ j=1,...,p E ũj , then on the one hand, x = (0, . . . , 0, x) with x ∈ R q-p . On the other hand, since also x ∈ E u j for j = p + 1, . . . , q, we must have 0 ∈ (0, x), u ′′ j u ′ 1 + (u ′ 1 ) ⊥ . Thus (0, x), u ′′ j = 0 and finally x = 0. Again, from Proposition 3.2 we get that (S 0 , H u

1 ∪ • • • ∪ H uq ∪ H ũ1 ∪ • • • ∪ H ũp ) is a
1 = s 2 ∈ R. Then (i) (S d-1 , H u,s 1 ∪ H u,s 2 ) is an AC(S d-1 )-Heisenberg uniqueness pair. (ii) (S d-1 , H u,s 1 ∪ H u,s 2 ) is a Heisenberg uniqueness pair if and only if |s 1 -s 2 | > π 2 . (iii) If s 1 , s 2 , s 3 are linearly independent over Q, then (S d-1 , H u,s 1 ∪ H u,s 2 ∪ H u,s 3 ) is a Heisenberg uniqueness pair.
The proof of (i) is essentially the same as in the case d = 2 by N. Lev [Le].

Proof. After a translation, it is enough to prove that (S d-1 , H u,-s ∪ H u,s ) is a Heisenberg uniqueness pair, where s = |s 1s 2 |/2. After applying a suitable rotation, we may further assume that u = (1, 0, . . . , 0). We will use the following notation: we write

x = (x 1 , x) for x ∈ R d . Let µ ∈ M(S d-1 ). According to Lemma 2.2, µ = 0 on H u,-s ∪H u,s is equiva- lent to R u * e ±isx 1 dµ(x 1 , x) = -e ±isx 1 dµ(x 1 , x). But R u * e ±isx 1 dµ(x 1 , x) = e ∓isx 1 dµ(-x 1 , x) thus (4.14) e +isx 1 dµ(x 1 , x) + e -isx 1 dµ(-x 1 , x) = 0 e -isx 1 dµ(x 1 , x) + e +isx 1 dµ(-x 1 , x) = 0 .
Writing each equation in the form e ±2isx 1 dµ(x 1 , x) = -dµ(-x 1 , x) and subtracting these equations, we obtain that sin(2sx Further,(4.14) shows that µ = 0 on

1 )dµ(x 1 , x) = 0. Conse- quently µ is supported on S d-1 ∩ π 2s Z × R d-1 .
S d-1 ∩ ({0} × R d-1 ) so that µ is actually supported on S d-1 ∩ π 2s Z * × R d-1
. Note that µ is supported on a set which has surface measure zero on S d-1 . Thus, if µ is absolutely continuous with respect to surface measure on S d-1 , then µ = 0 and (S d-1 , H u,-s ∪ H u,s ) is an AC(S d-1 )-Heisenberg uniqueness pair.

In the general case, µ =

k∈Z * δ π 2s k ⊗ µ k . Since supp µ ⊆ S d-1 , this representation of µ implies that µ k = 0 if π 2s |k| > 1 and that µ k is supported on 1 -π 2 4s 2 k 2 1/2 S d-2 if |k| ≤ 2 π s. Moreover, the symmetry (4.14) implies that e i π 2 k µ k + e -i π 2 k µ -k = 0, that is µ -k = (-1) k-1 µ k . In particular, if s > π
2 , then µ = δ 0 ⊗ µ 0 and thus µ = 0.

On the other hand, if s ≤ π 2 , we take the µ k 's to be arbitrary measures supported on 1 -π 2 4s 2 k 2 1/2 S d-2 for 0 < k ≤ 2 π s and then define µ

-k = (-1) k-1 µ k for those k's. The measure µ = 0<|k|≤ 2 π s δ π 2s k ⊗ µ k satisfies µ = 0 on H u,-s ∪ H u,s . Thus (S d-1 , H u,-s ∪ H u,s ) is not a Heisenberg uniqueness pair.
Note that in this case, µ(ξ 1 , ξ) = 0<|k|≤ 2 π s e -iπ 2s kξ 1 µ k ( ξ) is 4πs periodic in the first variable. Consequently, after a translation, if µ ∈ M(S d-1 ) and µ = 0 on H 0,s 1 ∪ H 0,s 2 , then µ is 2(s 2s 1 )π periodic in the first variable ξ 1 . Thus, if µ = 0 on H 0,s 3 as well, then µ is also 2(s 3s 1 )π periodic in ξ 1 . In particular, for every k, ℓ ∈ Z we have µ(s 1 + 2k(s 2s 1 )π + 2ℓ(s 3s 1 )π, ξ) = µ(s 1 , ξ) = 0. If s 1 , s 2 , s 3 are linearly independent over Q then the set {s 1 + 2k(s 2s 1 )π + 2ℓ(s 3s 1 )π : k, ℓ ∈ Z} is dense in R thus µ = 0 on a dense set in R d . As µ is continuous, µ = 0 on R d and thus µ = 0.

Let us now turn to N hyperplanes H 1 , . . . , H N that intersect in a common point. Without loss of generality, we will assume that H 1 , . . . , H N intersect in 0 so that H k = H u k ,0 for some u k ∈ R d , |u k | = 1. Let G be the group generated by the reflections {R 1 , . . . , R k }. This is a Coxeter group. The modulus |μ| is therefore invariant under the Coxeter group G = R 1 , . . . , R k . Let u, u ′ ∈ S d-1 and x ∈ H u . Since R u ′ x, R u ′ u = x, u , we have R u ′ H u = H R u ′ u . Consequently, if g = R j 1 R j 2 . . . R jn ∈ G and μ = 0 on H u , then μ vanishes on H gu . Furthermore note that the composition R u ′ R u of two reflections is a rotation in the plane spanned by u and u ′ with angle twice the angle between u and u ′ .

We distinguish two cases: (i) Either G is infinite. In this case [START_REF] Hohlweg | Ripoll Asymptotical behaviour of roots of infinite Coxeter groups[END_REF]Lemma 4.9], G contains a subgroup generated by two reflections R, R ′ that is already infinite. Write H u , H u ′ for the corresponding hyperplanes and note that u ′ / ∈ H u . But then the rotation RR ′ has an angle that is an irrational multiple of π, so that the orbit of H u under RR ′ is dense in R d . As µ = 0 on H u and | µ| is invariant under RR ′ , it follows that µ = 0 on a dense set. As µ is continuous, µ = 0 on R d thus µ = 0. Consequently, (S, H 1 ∪ • • • ∪ H N ) is a Heisenberg uniqueness pair.

(ii) Or G is finite. But then, there exists a subset W of R d (a Weyl chamber) such that {gW : g ∈ G} is a covering of R d and every x ∈ R d determines a unique g such that x ∈ gW . Now take any continuous function ϕ on S d-1 ∩ W and extend ϕ to S d-1 by the following rule: if x ∈ S d-1 then there exists a unique g ∈ G such that gx ∈ W . Writing g as g = M j=1 R k j with k 1 , . . . , k M ∈ {1, . . . , N}, we then set ϕ(x) = (-1) M ϕ(gx). The corresponding measure µ(x) = ϕ(x) dσ(x) is well defined, non-zero, absolutely continuous with respect to surface measure, and satisfies R k * µ = -µ by construction. By Lemma 2.2 µ = 0 on H 1 ∪ • • • ∪ H N . Thus (S, H 1 ∪ • • • ∪ H N ) fails to be a Heisenberg uniqueness pair.

Remark 2. When d = 2 we recover the theorem of Lev and Sjölin: let ℓ 1 , ℓ 2 be two lines that intersect at 0. Then (S 1 , ℓ 1 ∪ ℓ 2 ) is a Heisenberg uniqueness pair, if and only if the angle between ℓ 1 and ℓ 2 is not in Qπ.

However, when d ≥ 3, there exists sets of 3 hyperplanes such that the angle between any two of them is in Qπ but such that the corresponding Coxeter group is infinite, see [RS] for a complete description of the 3-dimensional case.

French State, managed by the French National Research Agency (ANR) in the frame of the Investments for the future Programme IdEx Bordeaux -CPU (ANR-10-IDEX-03-02). K. G. was supported in part by the project P26273 -N25 of the Austrian Science Fund (FWF). Both authors kindly acknowledge the support of the Austrian-French AMADEUS project 35598VB -ChargeDisq.

  Heisenberg uniqueness pair. 4. The sphere In this section Q(x) = |x| 2 , B(x, y) = x, y , v = 0 and S = S d-1 is the unit sphere of R d . If |u| = 1, R u is the reflection with respect to the hyperplane normal to u and R * u = R u . Let us first consider the case of parallel hyperplanes. Proposition 4.1. Let u ∈ S d-1 and s

  Theorem 4.2. Let H 1 , . . . , H N be hyperplanes in R d and R 1 , . . . , R N be the corresponding orthogonal reflections. Then(S d-1 , H 1 ∪ • • • ∪ H N ) is aHeisenberg uniqueness pair if and only if the Coxeter group generated by R 1 , . . . , R N is infinite. Proof. Let R k be the reflection with respect to H k . If µ ∈ M(S d-1 ) and µ vanishes on H k , then by Lemma 2.2 we have R k * µ = -µ. Consequently, its Fourier transform satisfies |μ(R k ξ)| = |μ(ξ)| for all ξ ∈ R d .

  and again j=1,...,p E u j ∩ j=1,...,p E ũj = {0}. In both cases, Proposition 3.2 shows that (S 0 , Hu 1 ∪• • •∪H up ∪H ũ1 ∪• • •∪H ũp )is a Heisenberg uniqueness pair. Thus (ii) is proved. Now, if ρ < 0 and q > p, then

	E u j ∩
	j=1,...,p	j=1,...,p

Acknowledgments

The authors kindly acknowledge financial support from the French ANR programs ANR 2011 BS01 007 01 (GeMeCod), ANR-12-BS01-0001 (Aventures). This study has been carried out with financial support from the

Note also that, if |u| = 1 and Q(u) = 0, then

, we thus obtain ũ, u = -1. An easy computation now shows that

3.1. Parallel hyperplanes with non-isotropic normal.

We will now prove the following statement.

Proposition 3.1. Let 1 ≤ p, q ≤ d -1 be two integers with p + q = d, s 1 , s 2 , s 3 ∈ R, and let

Proof. Translating the hyperplanes in the direction u, it is enough to consider the case s 1 = 0. We thus want to prove that (S ρ , H u,0 ∪ H u,s ∪ H u,t ) is a Heisenberg uniqueness pair when s, t are Q-independent.

From (3.10), we obtain that µ satisfies the following symmetries

It follows that, on the one hand, (3.11) µ(ξ + s 0 u) =µ R * u (ξ + s 0 u) =µ(-s 0 u + s 0 π u ⊥ ũ + ξ) ,