While both fundamental limits and system implementations are well understood for the point-to-point communication system, much less is developed for general communication networks. This thesis contrib...

We use Sidon sets to present an elementary method to study some combinatorial problems in finite fields, such as sum product estimates, solvability of some equations and the distribution of their solu...

We show that any connected Cayley graph Γ on an Abelian group of order 2n and degree Ω̃(log n) has at most 2(1+ o(1)) independent sets. This bound is tight up to to the o(1) term when Γ is bipartite. ...

We prove results about the L^p-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in L^p, and gives a very short pro...

We prove in particular that if A be a compact convex subset of R^n, and B from R^n be an arbitrary compact set then \mu (A-A) \ll \mu(A+B)^2 / (\sqrt{n} \mu (A)), provided that \mu(B)\ge \mu(A).

We propose a new model of a weakly random source that admits randomness extraction. Our model of additive sources includes such natural sources as uniform distributions on arithmetic progressions (APs...

We present the mathematical work of Yahya Ould Hamidoune, emphasizing his main achievements, notably in graph theory and additive combinatorics.

We present sum-set inequalities specialized to the generalized degrees of freedom (GDoF) framework. These are information theoretic lower bounds on the entropy of bounded density linear combinations o...

We present a short and self-contained proof of Jin's theorem about the piecewise syndeticity of difference sets which is entirely elementary, in the sense that no use is made of nonstandard analysis, ...

We present a new method to bound the cardinality of product sets in groups and give three applications. A new and unexpectedly short proof of the Plünnecke-Ruzsa sumset inequalities for commutative gr...

We investigate the lower asymptotic density of sumsets in $\mathbb{N}^2$ by proving certain Pl\"unnecke type inequalities for various notions of lower density in $\mathbb{N}^2$. More specifically, we ...

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptograph...

We establish several sum-product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)|+|AA| is substantially larger than |A| for an arbitrary polynomial f ov...

We consider the problem of secure distributed matrix multiplication (SDMM) in which a user wishes to compute the product of two matrices with the assistance of honest but curious servers. We construct...

Using the polynomial method in additive number theory, this article establishes a new addition theorem for the set of subsums of a set satisfying $A\cap(-A)=\emptyset$ in $\mathbb{Z}/p\mathbb{Z}$: \[|...

This text contains over three hundred specific open questions on various topics in additive combinatorics, each placed in context by reviewing all relevant results. While the primary purpose is to pro...

Suppose $G$ is a finite abelian group and $S$ is a sequence of elements in $G$. For any element $g$ of $G$, let $N_g(S)$ denote the number of subsequences of $S$ with sum $g$. The purpose of this pape...

Plunnecke's inequality is a standard tool for obtaining estimates on the cardinality of sumsets and has many applications in additive combinatorics. We present a new proof. The main novelty is that th...

One classical result of Freiman gives the optimal lower bound for the cardinality of $$A+A$$A+A if $$A$$A is a $$d$$d-dimensional finite set in $$\mathbb R^d$$Rd. Matolcsi and Ruzsa have recently gene...

Let n ≥ 2 be a fixed integer. Define (x)n to be the unique integer in the range 0 ≤ (x)n < n which is congruent to x modulo n. Given x1,…,xl ∈ ℤ, let ∥(x1,…,xl)∥1 = min{(ux1)n + ⋯ + (uxl)n:u ∈ ℤ, g......