Connections between covers of $\mathbb{Z}$ and subset sums

Abstract

In this paper we establish connections between covers of $\mathbb{Z}$ by residue classes and subset sums in a field. Suppose that $A_0=\{a_s(n_s){\}}_{s=0}^k$ covers each integer at least $p$ times with the residue class $a_0(n_0)=a_0+n_0\mathbb{Z}$ irredundant, where $p$ is a prime not dividing any of $n_1,\dots ,n_k$. Let $m_1,\dots ,m_k\in \mathbb{Z}$ be relatively prime to $n_1,\dots ,n_k$ respectively. For any $c,c_1,\dots ,c_k\in \mathbb{Z}/p\mathbb{Z}$ with $c_1\cdots c_k\ne 0$, we show that the set $$\bigg\{\bigg\{\sum_{s\in I}\frac{m_s}{n_s}\bigg\}:\, I\subseteq\{1,\ldots,k\} \ \mbox{and}\ \sum_{s\in I}c_s=c\bigg\}$$ contains an arithmetic progression of length $n_0$ with common difference $1/n_0$, where $\{x\}$ denotes the fractional part of a real number $x$.

17 pages

References in corpus (1)

• Discrete mathematics: methods and challenges