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A sequence of positive integers $(a_1,a_2,\ldots,a_k)$ is called $\ell$-additive if $a_1+a_2+\cdots+a_k=\ell a_1$ or $\ell a_k$. In this paper, we prove that for all $k\geq3$, if $n$ is sufficiently large, then every permutation of…

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any…

Non-uniquely decodable codes can be defined as the codes that cannot be uniquely decoded without additional disambiguation information. These are mainly the class of non-prefix-free codes, where a codeword can be a prefix of other(s), and…

This article extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a…

An $n \times m$ array with nonnegative entries is called doubly stochastic if the sum of its entries at each row is $m$ and at each column is $n$. The set of all $n \times m$ doubly stochastic arrays is a convex polytope with finitely many…

Let $A$ be a set of $n$ positive integers. We say that a subset $B$ of $A$ is a divisor of $A$, if the sum of the elements in $B$ divides the sum of the elements in $A$. We are interested in the following extremal problem. For each $n$,…

A set $A$ of nonnegative integers is an asymptotic basis of order $h$ if every sufficiently large integer can be represented as the sum of $h$ integers (not necessarily distinct) of $A$. An asymptotic basis $A$ of order $h$ is minimal if no…

We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solution in boxes with the side…

Given a finite set of bases $b_1$, $b_2$, \dots, $b_r$ (integers greater than $1$), a multi-base representation of an integer~$n$ is a sum with summands $db_1^{\alpha_1}b_2^{\alpha_2} \cdots b_r^{\alpha_r}$, where the $\alpha_j$ are…

A k-gap is a finite k-sequence of pairwise disjoint monotone families of infinite subsets of N mixed in such a way that we cannot find a partition of N such that each family is trival on one piece of the partition. We prove that, relative…

Let $G$ be an additive abelian group. Let $A=\{a_{0}, a_{1},\ldots, a_{k-1}\}$ be a nonempty finite subset of $G$. For a positive integer $h$ satisfying $1\leq h\leq k$, we let \[h\hat{}_{\underline{+}}A:=\{\Sigma_{i=0}^{k-1}\lambda_{i}…

Let $A$ be a nonempty finite subset of an additive abelian group $G$. Define $A + A := \{a + b : a, b \in A\}$ and $A \dotplus A := \{a + b : a, b \in A~\text{and}~ a \neq b\}$. The set $A$ is called a {\em sum-dominant (SD) set} if $|A +…

Restricted non-linear approximation is a type of N-term approximation where a measure $\nu$ on the index set (rather than the counting measure) is used to control the number of terms in the approximation. We show that embeddings for…

The definition of $k^{th}$-order empirical entropy of strings is extended to node labelled binary trees. A suitable binary encoding of tree straight-line programs (that have been used for grammar-based tree compression before) is shown to…

It is an open problem whether $ \binom{2n}{n} $ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $ m \nmid \binom{2^{k+1}}{2^{k}} $ is 0. To do…

Weird numbers are abundant numbers that are not pseudoperfect. Since their introduction, the existence of odd weird numbers has been an open problem. In this work, we describe our computational effort to search for odd weird numbers, which…

The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n$ rows and $n$ columns that avoids $P$. A 0-1 matrix $P$ is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for…

The no-(k+1)-in line problem seeks the maximum number of points that can be selected from an $n \times n$ square lattice such that no $k+1$ of them are collinear. The problem was first posed more than $100$ years ago for the special case…

A nonempty finite set of positive integers A is relatively prime if gcd(A) = 1 and it is relatively prime to n if gcd(A [ fng) = 1. The number of nonempty subsets of A which are relatively prime to n is \Phi(A, n) and the number of such…

We improve the estimates in the restriction problem in dimension $n \ge 4$. To do so, we establish a weak version of a $k$-linear restriction estimate for any $k$. The exponents in this weak $k$-linear estimate are sharp for all $k$ and…