Related papers: On various restricted sumsets

In this article, we first describe all nonempty sets of integers S with the property that for all n and m in S, not necessarily distinct, the set {n-m,n+m} intersected with S consists of a single element. These are the sets with at most two…

A tri-colored sum-free set in an abelian group $H$ is a collection of ordered triples in $H^3$, $\{(a_i,b_i,c_i)\}_{i=1}^m$, such that the equation $a_i+b_j+c_k=0$ holds if and only if $i=j=k$. Using a variant of the lemma introduced by…

The main interest of this article is the one-sided boundedness of the local discrepancy of $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ on the interval $(0,c)\subset(0,1)$ defined by \[D_n(\alpha,c)=\sum_{j=1}^n 1_{\{\{j\alpha\}<c\}}-cn.\] We…

This work describes the number of restricted finite words in the alphabet A={a,b} required to identify an infinite word with some period n in the set of all infinite words in this alphabet given up to a shift. Also reviewed the case of…

We derive upper and lower bounds on the sum of distances of a spherical code of size $N$ in $n$ dimensions when $N\sim n^\alpha, 0<\alpha\le 2.$ The bounds are derived by specializing recent general, universal bounds on energy of spherical…

We present a method for the enumeration of restricted words over a finite alphabet. Restrictions are described through the inclusion or exclusion of suitable building blocks used to construct the words by concatenation. Our approach, which…

Let $A(s) = \sum_n a_n n^{-s}$ be a Dirichlet series admitting meromorphic continuation to the complex plane. Assume we know the location of the poles of $A(s)$ with $|\Im s| \leq T$, and their residues, for some large constant $T$. It is…

For each $n, d \in \mathbb{N}$ and $0 < \alpha < 1$, we define a random subset of $\mathcal{A}^{\{1, 2, \dots, n\}^d}$ by independently including each element with probability $\alpha$ and excluding it with probability $1-\alpha$, and…

We introduce and study four optimization problems that generalize the well-known subset sum problem. Given a node-weighted digraph, select a subset of vertices whose total weight does not exceed a given budget. Some additional constraints…

The aim of this paper is to prove a general version of Pl\"unnecke's inequality. Namely, assume that for finite sets $A$, $B_1, ... B_k$ we have information on the size of the sumsets $A+B_{i_1}+... +B_{i_l}$ for all choices of indices…

Given $A$ a set of $N$ positive integers, an old question in additive combinatorics asks that whether $A$ contains a sum-free subset of size at least $N/3+\omega(N)$ for some increasing unbounded function $\omega$. The question is generally…

We address the "sums of dilates problem" by looking for non-trivial lower bounds on sumsets of the form $k \cdot X + l \cdot X$, where $k$ and $l$ are non-zero integers and $X$ is a subset of a possibly non-abelian group $G$ (written…

We extend local limit theorem type results to nonconventional sums of the form $S_N=\sum_{n=1}^NF(\xi_n,\xi_{2n},...,\xi_{\ell n})$.

Let S be an abelian semigroup, written additively. Let A be a finite subset of S. We denote the cardinality of A by |A|. For any positive integer h, the sumset hA is the set of all sums of h not necessarily distinct elements of A. We define…

For a vector $\mathbf a=(a_1,\ldots,a_r)$ of positive integers we prove formulas for the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0,…

Let f(x_1,x_2,...,x_m) = u_1x_1+u_2 x_2+... + u_mx_m be a linear form with positive integer coefficients, and let N_f(k) = min{|f(A)| : A \subseteq Z and |A|=k}. A minimizing k-set for f is a set A such that |A|=k and |f(A)| = N_f(k). A…

We find the number of compositions over finite abelian groups under two types of restrictions: (i) each part belongs to a given subset and (ii) small runs of consecutive parts must have given properties. Waring's problem over finite fields…

Given a finite subset S in F_p^d, let a(S) be the number of distinct r-tuples (x_1,...,x_r) in S such that x_1+...+x_r = 0. We consider the "moments" F(m,n) = sum_|S|=n a(S)^m. Specifically, we present an explicit formula for F(m,n) as a…

Define $S_n^k(\alpha)$ to be the set of permutations of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $\alpha \in S_m$. Let $s_n^k(\alpha)$ be the size of $S_n^k(\alpha)$. We investigate $S_n^0(\alpha)$ for all…

New lower bounds involving sum, difference, product, and ratio sets for a set $A\subset \C$ are given. The estimates involving the sum set match, up to constants, the one obtained by Solymosi for the reals and are obtained by generalising…