A Littlewood-Offord kind of problem in $\mathbb{Z}_p$ and $\Gamma$-sequenceability

Abstract

The Littlewood-Offord problem is a classical question in probability theory and discrete mathematics, proposed, firstly by Littlewood and Offord in the 1940s. Given a set $A$ of integer, this problem asks for an upper bound on the probability that a randomly chosen subset $X$ of $A$ sums to an integer $x$. This article proposes a variation of the problem, considering a subset $A$ of a cyclic group of prime order and examining subsets $X\subseteq A$ of a given cardinality $\ell$. The main focus of this paper is then on bounding the probability distribution of the sum $Y$ of $\ell$ i.i.d. $Y_1,\dots, Y_{\ell}$ whose support is contained in $\mathbb{Z}_p$. The main result here presented is that, if the probability distributions of the variables $Y_i$ are bounded by $\lambda \leq 9/10$, then, assuming that $p> \frac{2}{\lambda}\left(\frac{\ell_0}{3}\right)^{\nu}$ (for some $\ell_0\leq\ell$), the distribution of $Y$ is bounded by $\lambda\left(\frac{3}{\ell_0}\right)^{\nu}$ for some positive absolute constant $\nu$. Then an analogous result is implied for the Littlewood-Offord problem over $\mathbb{Z}_p$ on subsets $X$ of a given cardinality $\ell$ in the regime where $n$ is large enough. Finally, as an application of our results, we propose a variation of the set-sequenceability problem: that of $\Gamma$-sequenceability. Given a graph $\Gamma$ on the vertex set $\{1,2,\dots,n\}$ and given a subset $A\subseteq \mathbb{Z}_p$ of size $n$, here we want to find an ordering of $A$ such that the partial sums $s_i$ and $s_j$ are different whenever $\{i,j\}\in E(\Gamma)$. As a consequence of our results on the Littlewood-Offord problem, we have been able to prove that, if the maximum degree of $\Gamma$ is at most $d$, $n$ is large enough, and $p>n^2$, any subset $A\subseteq \mathbb{Z}_p$ of size $n$ is $\Gamma$-sequenceable.