Related papers: On an inverse problem for restricted sumsets

Let A be a finite nonempty subset of an additive abelian group G, and let \Sigma(A) denote the set of all group elements representable as a sum of some subset of A. We prove that |\Sigma(A)| >= |H| + 1/64 |A H|^2 where H is the stabilizer…

Let $h \geq 2$, $k \geq 5$ be integers and $A$ be a nonempty finite set of $k$ integers. Very recently, Tang and Xing studied extended inverse theorems for $hk-h+1 < \left|hA\right| \leq hk+2h-3$. In this paper, we extend the work of Tang…

Suppose that $m,n\in \mathbb{N}$ and that $A:\mathbb{R}^m\to \mathbb{R}^n$ is a linear operator. It is shown here that if $k,r\in \mathbb{N}$ satisfy $k<r\le \mathrm{\bf rank(A)}$ then there exists a subset $\sigma\subseteq \{1,\ldots,m\}$…

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 $k \ge 2$ and $s$ be positive integers, and let $n$ be a large positive integer subject to certain local conditions. We prove that if $s \ge k^2+k+1$ and $\theta > 31/40$, then $n$ can be expressed as a sum $p_1^k + \dots + p_s^k$,…

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…

Let Sym_n denote the symmetric group of all permutations pi = a_1...a_n of {1,...,n}. An index i is a peak of pi if a_{i-1} < a_i > a_{i+1} and we let P(pi) be the set of peaks of pi. Given any set S of positive integers we define P(S;n) to…

Let $k$ and $n$ be positive integers, $n>k$. Define $r(n,k)$ to be the minimum positive value of $$ |\sqrt{a_1} + ... + \sqrt{a_k} - \sqrt{b_1} - >... -\sqrt{b_k} | $$ where $ a_1, a_2, ..., a_k, b_1, b_2, ..., b_k $ are positive integers…

In this paper we present three different results dealing with the number of $(\leq k)$-facets of a set of points: 1. We give structural properties of sets in the plane that achieve the optimal lower bound $3\binom{k+2}{2}$ of $(\leq…

In this paper we prove that every set $A\subset\mathbb{Z}$ satisfying the inequality $\sum_{x}\min(1_A*1_A(x),t)\le(2+\delta)t|A|$ for $t$ and $\delta$ in suitable ranges, then $A$ must be very close to an arithmetic progression. We use…

Let $f$ be a smooth real function with strictly monotone first $k$ derivatives. We show that for a finite set $A$, with $|A+A|\leq K|A|$, $|2^kf(A)-(2^k-1)f(A)|\gg_k |A|^{k+1-o(1)}/K^{O_k(1)}$. We deduce several new sum-product type…

The following is shown : Let $S=\{a_1,a_2,..,a_{2n}\}$ be a subset of a totally ordered commutative semi-group $(G,*,\leq)$ with $a_1\leq a_2\leq...\leq a_{2n}$. Provided that a system of $n$ $a_{i_k} * a_{j_k}\ (a_{i_k}, a_{j_k} \in G ;\ 1…

Let $G=C_n\oplus C_n$ and let $k\in [0,n-1]$. We study the structure of sequences of terms from $G$ with maximal length $|S|=2n-2+k$ that fail to contain a nontrivial zero-sum subsequence of length at most $2n-1-k$. For $k\leq 1$, this is…

A set of integers is called sum-free if it contains no triple $(x,y,z)$ of not necessarily distinct elements with $x+y=z$. In this paper, we provide a structural characterisation of sum-free subsets of $\{1,2,\ldots,n\}$ of density at least…

We present a variety of new results on finite sets A of integers for which the sumset A+A is larger than the difference set A-A, so-called MSTD (more sums than differences) sets. First we show that there is, up to affine transformation, a…

By making use of arithmetic information inequalities, we give a strong quantitative bound for the discretised ring theorem. In particular, we show that if $A \subset [1,2]$ is a $(\delta,\sigma)$-set, with $|A| = \delta^{-\sigma},$ then…

We introduce the "Median Inverse Problem" for metric spaces. In particular, having a permutation $\pi$ in the symmetric group $S_n$ (endowed with the breakpoint distance), we study the set of all $k$-subsets $\{x_1,...,x_k\}\subset S_n$ for…

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…

It is known that for an arbitrary positive integer \(n\) the sequence \(S(x^n)=(1^n, 2^n, \ldots)\) is complete, meaning that every sufficiently large integer is a sum of distinct \(n\)th powers of positive integers. We prove that every…

Suppose that we have a set of numbers x_1, ..., x_n which have nonnegative sum. How many subsets of k numbers from {x_1, ..., x_n} must have nonnegative sum? Manickam, Miklos, and Singhi conjectured that for n at least 4k the answer is (n-1…