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- W4385932289 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 $Xsubseteq A$ 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 $ell_0leqell$), the distribution of $Y$ is bounded by $lambdaleft(frac{3}{ell_0}right)^{nu}$ 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 $Asubseteq mathbb{Z}_p$ 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 $Asubseteq mathbb{Z}_p$ of size $n$ is $Gamma$-sequenceable." @default.
- W4385932289 created "2023-08-18" @default.
- W4385932289 creator A5081717981 @default.
- W4385932289 date "2023-08-08" @default.
- W4385932289 modified "2023-09-27" @default.
- W4385932289 title "A Littlewood-Offord kind of problem in $mathbb{Z}_p$ and $Gamma$-sequenceability" @default.
- W4385932289 doi "https://doi.org/10.48550/arxiv.2308.04284" @default.
- W4385932289 hasPublicationYear "2023" @default.
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