Comparison estimates for linear forms in additive number theory

Abstract

Let $R$ be a commutative ring $R$ with $1_R$ and with group of units $R^{\times}$. Let $\Phi = \Phi(t_1,\ldots, t_h) = \sum_{i=1}^h \varphi_it_i$ be an $h$-ary linear form with nonzero coefficients $\varphi_1,\ldots, \varphi_h \in R$. Let $M$ be an $R$-module. For every subset $A$ of $M$, the image of $A$ under $\Phi$ is \[ \Phi(A) = \{ \Phi(a_1,\ldots, a_h) : (a_1,\ldots, a_h) \in A^h \}. \] For every subset $I$ of $\{1,2,\ldots, h\}$, there is the subset sum $ s_I = \sum_{i\in I} \varphi_i. $ Let $ \mathcal{S} (\Phi) = \{s_I: \emptyset \neq I \subseteq \{1,2,\ldots, h\} \}. $ Theorem. Let $\Upsilon(t_1,\ldots, t_g) = \sum_{i=1}^g \upsilon_it_i$ and $\Phi(t_1,\ldots, t_h) = \sum_{i=1}^h \varphi_it_i$ be linear forms with nonzero coefficients in the ring $R$. If $\{0, 1\} \subseteq \mathcal{S} (\Upsilon)$ and $\mathcal{S} (\Phi) \subseteq R^{\times}$, then for every $\varepsilon > 0$ and $c > 1$ there exist a finite $R$-module $M$ with $|M| > c$ and a subset $A$ of $M$ such that $\Upsilon(A \cup \{0\}) = M$ and $|\Phi(A)| < \varepsilon |M|$.