arXiv subjects

Alexander Fish

Publications and source records attributed to Alexander Fish.

At least 37 records · Page 2Linked to original sources

arXiv2019

Sets of transfer times with small densities

We consider in this paper the set of transfer times between two measurable subsets of positive measures in an ergodic probability measure-preserving system of a countable abelian group. If the lower asymptotic density of the transfer times is small, then we prove this set must be either periodic or Sturmian. Our results can be viewed as ergodic-theoretical extensions of some classical sumset theorems in compact abelian groups due to Kneser. Our proofs are based on a correspondence principle for action sets which was developed previously by the first two authors.

math.DS↗

arXiv2019

Approximate invariance for ergodic actions of amenable groups

We develop in this paper some general techniques to analyze action sets of small doubling for probability measure-preserving actions of amenable groups. As an application of these techniques, we prove a dynamical generalization of Kneser's celebrated density theorem for subsets in $(\bZ,+)$, valid for any countable amenable group, and we show how it can be used to establish a plethora of new inverse product set theorems for upper and lower asymptotic densities. We provide several examples demonstrating that our results are optimal for the settings under study.

math.DS↗

arXiv2019

Extensions of Schreiber's theorem on discrete approximate subgroups in $\mathbb{R}^d$

In this paper we give an alternative proof of Schreiber's theorem which says that an infinite discrete approximate subgroup in $\mathbb{R}^d$ is relatively dense around a subspace. We also deduce from Schreiber's theorem two new results. The first one says that any infinite discrete approximate subgroup in $\mathbb{R}^d$ is a restriction of a Meyer set to a thickening of a linear subspace in $\mathbb{R}^d$, and the second one provides an extension of Schreiber's theorem to the case of the Heisenberg group.

math.DS↗

arXiv2018

Delocalisation of one-dimensional marginals of product measures and the capacity of LTI discrete channels

We consider discrete linear time invariant (LTI) channels satisfying the phase independence (PI) assumption. We show that under the PI assumption the capacity of LTI channels is positive. The main technical tool that we use to establish the positivity of the capacity is the delocalisation theorem for one-dimensional marginals of the product measure due to Ball and Nazarov. We also prove two delocalisation results that can be seen as extensions of Ball-Nazarov Theorem.

cs.IT↗

arXiv2017

Twisted Recurrence via Polynomial Walks

In this paper we show how polynomial walks can be used to establish a twisted recurrence for sets of positive density in $\mathbb{Z}^d$. In particular, we prove that if $Γ\leq \operatorname{GL}_d(\mathbb{Z})$ is finitely generated by unipotents and acts irreducibly on $\mathbb{R}^d$, then for any set $B \subset \mathbb{Z}^d$ of positive density, there exists $k \geq 1$ such that for any $v \in k \mathbb{Z}^d$ one can find $γ\in Γ$ with $γv \in B - B$. Our method does not require the linearity of the action, and we prove a twisted recurrence for semigroups of maps from $\mathbb{Z}^d$ to $\mathbb{Z}^d$ satisfying some irreducibility and polynomial assumptions. As one of the consequences, we prove a non-linear analog of Bogolubov's theorem -- for any set $B \subset \mathbb{Z}^2$ of positive density, and $p(n) \in \mathbb{Z}[n]$, with $p(0) = 0$ and $\operatorname{deg}(p) \geq 2$, there exists $k \geq 1$ such that $k \mathbb{Z} \subset \{ x - p(y) \, | \, (x,y) \in B-B \}$. Unlike the previous works on twisted recurrence that used recent results of Benoist-Quint and Bourgain-Furman-Lindenstrauss-Mozes on equidistribution of random walks on automorphism groups of tori, our method relies on the classical Weyl equidistribution for polynomial orbits on tori.

math.DS↗

arXiv2017

On Bohr sets of integer valued traceless matrices

In this paper we show that any Bohr-zero non-periodic set $B$ of traceless integer valued matrices, denoted by $Λ$, intersects non-trivially the conjugacy class of any matrix from $Λ$. As a corollary, we obtain that the family of characteristic polynomials of $B$ contains all characteristic polynomials of matrices from $Λ$. The main ingredient used in this paper is an equidistribution result for an $SL_d(\mathbf{Z})$ random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work.

math.DS↗

arXiv2017

On product of difference sets for sets of positive density

In this paper we prove that given two sets $E_1,E_2 \subset \mathbb{Z}$ of positive density, there exists $k \geq 1$ which is bounded by a number depending only on the densities of $E_1$ and $E_2$ such that $k\mathbb{Z} \subset (E_1-E_1)\cdot(E_2-E_2)$. As a corollary of the main theorem we deduce that if $α,β> 0$ then there exist $N_0$ and $d_0$ which depend only on $α$ and $β$ such that for every $N \geq N_0$ and $E_1,E_2 \subset \mathbb{Z}_N$ with $|E_1| \geq αN, |E_2| \geq βN$ there exists $d \leq d_0$ a divisor of $N$ satisfying $d \, \mathbb{Z}_N \subset (E_1-E_1)\cdot(E_2-E_2)$.

math.DS↗

arXiv2016

Unwinding spirals

We show that there is no bi-Lipschitz homeomorphism of $\mathbb{R}^2$ that maps a spiral with a sub-exponential decay of winding radii to an unwinded arc. This result is sharp as shows an example of a logarithmic spiral.

math.GT↗

arXiv2015

Characteristic Polynomial Patterns in Difference Sets of Matrices

We show that for every subset $E$ of positive density in the set of integer square-matrices with zero traces, there exists an integer $k \geq 1$ such that the set of characteristic polynomials of matrices in $E-E$ contains the set of \emph{all} characteristic polynomials of integer matrices with zero traces and entries divisible by $k$. Our theorem is derived from results by Benoist-Quint on measure rigidity for actions on homogeneous spaces.

math.DS↗

arXiv2015

Product set phenomena for countable groups

We develop in this paper general techniques to analyze local combinatorial structures in product sets of two subsets of a countable group which are "large" with respect to certain classes of (not necessarily invariant) means on the group. As applications of our methods, we extend and quantify a series of recent results by Jin, Bergelson-Furstenberg-Weiss, Beiglböck-Bergelson-Fish, Griesmer and DiNasso-Lupini to general countable groups.

math.DS↗

arXiv2014

Ergodic Theorems for coset spaces

We study in this paper the validity of the mean ergodic theorem along \emph{left} Følner sequences in a countable amenable group $G$. Although the \emph{weak} ergodic theorem always holds along \emph{any} left Følner sequence in $G$, we provide examples where the \emph{mean} ergodic theorem fails in quite dramatic ways. On the other hand, if $G$ does not admit any ICC quotients, e.g. if $G$ is virtually nilpotent, then we prove that the mean ergodic theorem does indeed hold along \emph{any} left Følner sequence. In the case when a unitary representation of a countable amenable group is induced from a unitary representation of a "sufficiently thin" subgroup, we prove that the mean ergodic theorem holds along any left Følner sequence for this representation. Furthermore, we show that every countable (infinite) amenable group $L$ embeds into a countable group $G$ which admits a unitary representation with the property that for any left Følner sequence $(F_n)$ in $L$, there exists a sequence $(s_n)$ in $G$ such that the mean (but \emph{not} the weak) ergodic theorem fails for this representation along the sequence $(F_n s_n)$. Finally, we provide examples of countable (not necessarily amenable) groups $G$ with proper, infinite-index subgroups $H$, so that the \emph{pointwise} ergodic theorem holds for averages along \emph{any} strictly increasing and nested sequence of finite subsets of the coset $G/H$.

math.DS↗

arXiv2014

Plünnecke inequalities for measure graphs with applications

We generalize Petridis's new proof of Plünnecke's graph inequality to graphs whose vertex set is a measure space. Consequently, this gives new Plünnecke inequalities for measure preserving actions which enable us to deduce, via a Furstenberg correspondence principle, Banach density estimates in countable abelian groups that improve on those given by Jin.

math.DS↗

arXiv2014

Performance Estimates of the Pseudo-Random Method for Radar Detection

A performance of the pseudo-random method for the radar detection is analyzed. The radar sends a pseudo-random sequence of length $N$, and receives echo from $r$ targets. We assume the natural assumptions of uniformity on the channel and of the square root cancellation on the noise. Then for $r \leq N^{1-δ}$, where $δ> 0$, the following holds: (i) the probability of detection goes to one, and (ii) the expected number of false targets goes to zero, as $N$ goes to infinity.

cs.IT↗

arXiv2013

Plünnecke inequalities for countable abelian groups

We establish in this paper a new form of Plünnecke-type inequalities for ergodic probability measure-preserving actions of any countable abelian group. Using a correspondence principle for product sets, this allows us to deduce lower bounds on the upper and lower Banach densities of any product set in terms of the upper Banach density of an iterated product set of one of its addends. These bounds are new already in the case of the integers. We also introduce the notion of an ergodic basis, which is parallel, but significantly weaker than the analogous notion of an additive basis, and deduce Plünnecke bounds on their impact functions with respect to both the upper and lower Banach densities on any countable abelian group.

math.DS↗

arXiv2013

Almost Linear Complexity Methods for Delay-Doppler Channel Estimation

A fundamental task in wireless communication is channel estimation: Compute the channel parameters a signal undergoes while traveling from a transmitter to a receiver. In the case of delay-Doppler channel, i.e., a signal undergoes only delay and Doppler shifts, a widely used method to compute delay-Doppler parameters is the pseudo-random method. It uses a pseudo-random sequence of length N; and, in case of non-trivial relative velocity between transmitter and receiver, its computational complexity is O(N^2logN) arithmetic operations. In [1] the flag method was introduced to provide a faster algorithm for delay-Doppler channel estimation. It uses specially designed flag sequences and its complexity is O(rNlogN) for channels of sparsity r. In these notes, we introduce the incidence and cross methods for channel estimation. They use triple-chirp and double-chirp sequences of length N, correspondingly. These sequences are closely related to chirp sequences widely used in radar systems. The arithmetic complexity of the incidence and cross methods is O(NlogN + r^3), and O(NlogN + r^2), respectively.

cs.IT↗

arXiv2013

The Incidence and Cross Methods for Efficient Radar Detection

The designation of the radar system is to detect the position and velocity of targets around us. The radar transmits a waveform, which is reflected back from the targets, and echo waveform is received. In a commonly used model, the echo is a sum of a superposition of several delay-Doppler shifts of the transmitted waveform, and a noise component. The delay and Doppler parameters encode, respectively, the distances, and relative velocities, between the targets and the radar. Using standard digital-to-analog and sampling techniques, the estimation task of the delay-Doppler parameters, which involves waveforms, is reduced to a problem for complex sequences of finite length N. In these notes we introduce the Incidence and Cross methods for radar detection. One of their advantages, is robustness to inhomogeneous radar scene, i.e., for sensing small targets in the vicinity of large objects. The arithmetic complexity of the incidence and cross methods is O(NlogN + r^3) and O(NlogN + r^2), for r targets, respectively. In the case of noisy environment, these are the fastest radar detection techniques. Both methods employ chirp sequences, which are commonly used by radar systems, and hence are attractive for real world applications.

cs.IT↗

arXiv2012

Delay-Doppler Channel Estimation with Almost Linear Complexity

A fundamental task in wireless communication is Channel Estimation: Compute the channel parameters a signal undergoes while traveling from a transmitter to a receiver. In the case of delay-Doppler channel, a widely used method is the Matched Filter algorithm. It uses a pseudo-random sequence of length N, and, in case of non-trivial relative velocity between transmitter and receiver, its computational complexity is O(N^{2}log(N)). In this paper we introduce a novel approach of designing sequences that allow faster channel estimation. Using group representation techniques we construct sequences, which enable us to introduce a new algorithm, called the flag method, that significantly improves the matched filter algorithm. The flag method finds the channel parameters in O(mNlog(N)) operations, for channel of sparsity m. We discuss applications of the flag method to GPS, radar system, and mobile communication as well.

cs.IT↗

arXiv2011

Computing the Matched Filter in Linear Time

A fundamental problem in wireless communication is the time-frequency shift (TFS) problem: Find the time-frequency shift of a signal in a noisy environment. The shift is the result of time asynchronization of a sender with a receiver, and of non-zero speed of a sender with respect to a receiver. A classical solution of a discrete analog of the TFS problem is called the matched filter algorithm. It uses a pseudo-random waveform S(t) of the length p, and its arithmetic complexity is O(p^{2} \cdot log (p)), using fast Fourier transform. In these notes we introduce a novel approach of designing new waveforms that allow faster matched filter algorithm. We use techniques from group representation theory to design waveforms S(t), which enable us to introduce two fast matched filter (FMF) algorithms, called the flag algorithm, and the cross algorithm. These methods solve the TFS problem in O(p\cdot log (p)) operations. We discuss applications of the algorithms to mobile communication, GPS, and radar.

cs.IT↗