Related papers: Sumsets in the Hypercube

Let $\mathcal{F}$ be an antichain of finite subsets of $\mathbb{N}$. How quickly can the quantities $|\mathcal{F}\cap 2^{[n]}|$ grow as $n\to\infty$? We show that for any sequence $(f_n)_{n\ge n_0}$ of positive integers satisfying…

A set of reals $A=\{a_1,...,a_n\}$ labeled in increasing order is called convex if there exists a continuous strictly convex function $f$ such that $f(i)=a_i$ for every $i$. Given a convex set $A$, we prove…

A family $\mathcal{F}\subset 2^G$ of subsets of an abelian group $G$ is a Sidon system if the sumsets $A+B$ with $A,B\in \mathcal{F}$ are pairwise distinct. Cilleruelo, Serra and the author previously proved that the maximum size $F_k(n)$…

In the paper we are studying some properties of subsets Q of sums of dissociated sets. The exact upper bound for the number of solutions of the following equation (1) q_1 + ... + q_p = q_{p+1} + ... + q_{2p}, q_i \in Q in groups F_2^n is…

A set of integers is sum-free if it contains no solution to the equation $x+y=z$. We study sum-free subsets of the set of integers $[n]=\{1,\ldots,n\}$ for which the integer $2n+1$ cannot be represented as a sum of their elements. We prove…

Let $G$ be an induced subgraph of the hypercube $Q_k$ for some $k$. We show that if $|G|$ is a power of $2$ then, for sufficiciently large $n$, the vertex set of $Q_n$ can be partitioned into induced copies of $G$. This answers a question…

Let $G$ be a finite abelian group, and let $S$ be a sequence over $G$. Let $f(S)$ denote the number of elements in $G$ which can be expressed as the sum over a nonempty subsequence of $S$. In this paper, we determine all the sequences $S$…

{The first version of this text was written and submitted to a journal on April, 12, 2018. This second version was submitted on April, 9, 2019.} We investigate the existence of subsets $A$ and $B$ of $\mathbb{N}:=\{0,1,2,\dots\}$ such that…

A finite set of integers $A$ is a sum-dominant (also called an More Sums Than Differences or MSTD) set if $|A+A| > |A-A|$. While almost all subsets of $\{0, \dots, n\}$ are not sum-dominant, interestingly a small positive percentage are. We…

The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach's conjecture and Fermat's Last theorem) can be formulated in terms of the sumset $S + S = \{x+y :…

We are looking for integer sets that resemble classical Cantor set and investigate the structure of their sum sets. Especially we investigate $FS(B)$ the subset sum of sequence type $B=\{\lfloor p^n\alpha\rfloor\}^\infty_{n=0}$. When $p=2$,…

We study the intersection statistics of affine subspaces in the hypercube $\mathbb{F}_2^n$, motivated by recent work of Alon, Axenovich, and Goldwasser on the intersection statistics of axis-aligned subcubes of an $n$-dimensional cube. Let…

In this article we aim to develop from first principles a theory of sum sets and partial sum sets, which are defined analogously to difference sets and partial difference sets. We obtain non-existence results and characterisations. In…

The special case of the hypergeometric function $_{2}F_{1}$ represents the binomial series $(1+x)^{\alpha}=\sum_{n=0}^{\infty}(\:\alpha n\:)x^{n}$ that always converges when $|x|<1$. Convergence of the series at the endpoints, $x=\pm 1$,…

Two subsets $S$ and $T$ of $\mathbb{F}_2^n$ are \textit{affinely equivalent} if there is an affine automorphism of $\mathbb{F}_2^n$ taking $S$ to $T$. Given a basis of the affine span of $S$, we can construct a Venn diagram whose regions…

It is a classical fact that every $n$-element set of positive reals has at least $\binom{n+1}{2}+1$ distinct subset sums, with equality exactly for homogeneous arithmetic progressions (when $n\geq 4$). We establish stability versions of…

We consider several families of binomial sum identities whose definition involves the absolute value function. In particular, we consider centered double sums of the form \[S_{\alpha,\beta}(n) :=…

For a finite set $A\subset\mathbb{N}$ and $k\in \mathbb{N}$, let $\omega_k(A) = \sum_{i\in A, i\neq k}1$. For each $n\in \mathbb{N}$, define $$a_{k, n}\ =\ |\{E\subset \mathbb{N}\,:\, E = \emptyset\mbox{ or } \omega_k(E) < \min E\leqslant…

Let $\mathcal{A}$ be a finite set of integers, and let $h\mathcal{A}$ denote the $h$-fold sumset of $\mathcal{A}$. Let $(h\mathcal{A})^{(t)}$ be subset of $h\mathcal{A}$ consisting of all integers that have at least $t$ representations as a…

The $h$-fold sumset of a set $A$ of integers is the set of all sums of $h$ not necessarily distinct elements of $A$. Let $(A_q)_{q=1}^{\infty}$ be a strictly decreasing sequence of sets of integers and let $A = \bigcap_{q=1}^{\infty} A_q$.…