Linear forms and complementing sets of integers

Abstract

Let $\phi (x_1,\dots ,x_h,y)=u_1{x}_1+\cdots +u_h{x}_h+vy$ be a linear form with nonzero integer coefficients $u_1,\dots ,u_h,v.$ Let $\mathcal{A}=(A_1,\dots ,A_h)$ 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 $\phi$ and the sets \mca\ and $B$ as follows: $$R_{\mathcal{A},B}^{(\phi )}(n)=\text{card}\left(\left\{(a_1,\dots ,a_h,b)\in A_1\times \cdots \times A_h\times B:\phi (a_1,\dots ,a_h,b)=n\right\}\right).$$ 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 $\{\phi (a_1,\dots ,a_h,0):(a_1,\dots ,a_h)\in A_1\times \cdots \times A_h\}.$

10 pages