Connections between covers of $\mathbb Z$ and subset sums (2008.04153v1)

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 $n_1,\ldots,n_k$. Let $m_1,\ldots,m_k\in\mathbb Z$ be relatively prime to $n_1,\ldots,n_k$ respectively. For any $c,c_1,\ldots,c_k\in\mathbb Z/p\mathbb Z$ with $c_1\cdots c_k\not=0$, 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 $x$.