Pl\"unnecke inequalities for measure graphs with applications

We generalize Petridis's new proof of Pl\"unnecke's graph inequality to graphs whose vertex set is a measure space. Consequently, this gives new Pl\"unnecke 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.Comment: 24 pages, 1 figur

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.Comment: 4 double column pages. arXiv admin note: substantial text overlap with arXiv:1309.372

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