Comparison estimates for linear forms in additive number theory

TL;DR

This paper investigates the behavior of linear forms over modules in additive number theory, establishing conditions under which certain images cover entire modules while others remain small.

Contribution

It introduces new estimates for the images of subsets under linear forms, revealing how specific algebraic conditions influence their size and distribution.

Findings

Existence of modules where one linear form's image covers the entire module.

Construction of subsets with small image under another linear form.

Conditions linking sumsets and invertibility in the ring.

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 $|\Phi(A)| < \varepsilon |M|$.