Linear forms and complementing sets of integers

Melvyn B. Nathanson
Arxiv ID: 0801.0001Last updated: 12/30/2021
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 $\{ \varphi(a_1,\ldots,a_h,0): (a_1,\ldots, a_h) \in A_1 \times \cdots \times A_h\}.$

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