Bounded asymptotic bases for linear forms

Abstract

For a vector of positive integers $\mathbf{b} = (b_1,\ldots,b_h)$ with $\gcd(b_1,\ldots,b_h) = 1$, we study sets $A \subseteq \mathbb{N}$ for which every sufficiently large integer has a bounded positive number of representations \[ n = b_1 x_1 + \cdots + b_h x_h \qquad (x_1,\ldots,x_h\in A). \] We prove that such a set exists for every binary vector $\mathbf{b} \neq (1,1)$, and for some general higher-dimensional families, including $\mathbf{b} = (u_1, p^d u_2, \ldots, p^{(h-1)d} u_h)$ where $p\nmid u_1\cdots u_h$.