Comparison estimates for linear forms in additive number theory

Melvyn B. Nathanson

Published 2016-04-29Version 1

Let $M$ be an $R$-module. Consider the $h$-ary linear form $\Phi:M^h \rightarrow M$ defined by $\Phi(t_1,\ldots, t_h) = \sum_{i=1}^h \varphi_it_i$with nonzero coefficient sequence $(\varphi_1,\ldots, \varphi_h) \in R^h$. For every subset $A$ of $M$, define \[ \Phi(A) = \{ \Phi(a_1,\ldots, a_h) : (a_1,\ldots, a_h) \in A^h \}. \] For every nonempty subset $I$ of $\{1,2,\ldots, h\}$, define the subsequence sum $s_I = \sum_{i\in I} \varphi_i$. Let $\mathcal{S}(\Phi) = \{s_I: I \subseteq \{1,2,\ldots, h\}, I \neq \emptyset \} $ be the set of all nonempty subsequence sums of the sequence of coefficients of $\Phi$. Let $R^{\times}$ be the group of units of the ring $R$. 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 exists a finite $R$-module $M$ with $|M| > c$ and a subset $A$ of $M$ such that $\Upsilon(A \cup \{0\}) = M$ and $|\Phi(A)| < \varepsilon |M|$.