Linear forms and complementing sets of integers

TL;DR

This paper investigates the structure of integer sets related to linear forms, showing that if a certain representation function is constant, then the infinite set involved must be periodic with a bounded period.

Contribution

It establishes a link between constant representation functions of linear forms and the periodicity of the associated infinite set, providing bounds on the period based on the finite sets involved.

Findings

The representation function being constant implies the set B is periodic.

The period of B is bounded by the diameter of a related finite set.

The results connect linear forms with the structure of complementing sets.

Abstract

Let $\varphi(x_1,\ldots,x_h,y) = u_1x_1 + \cdots + u_hx_h+vy$ be a linear form with nonzero integer coefficients $u_1,\ldots, u_h, v.$ Let $\mathcal{A} = (A_1,\ldots, A_h)$ be an $h$-tuple of finite sets of integers and let $B$ be an infinite set of integers. Define the representation function associated to the form $\varphi$ and the sets \mca\ and $B$ as follows: $$ R^{(\varphi)}_{\mathcal{A},B}(n) = \text{card}\left( \left\{ (a_1,\ldots, a_h,b) \in A_1 \times \cdots \times A_h \times B: \varphi(a_1, \ldots , a_h,b ) = n \right\} \right).$$ If this representation function is constant, then the set $B$ is periodic and the period of $B$ will be bounded in terms of the diameter of the finite set $\{ \varphi(a_1,\ldots,a_h,0): (a_1,\ldots, a_h) \in A_1 \times \cdots \times A_h\}.$