Induced arithmetic removal: complexity 1 patterns over finite fields

TL;DR

This paper establishes an arithmetic analog of the induced graph removal lemma for complexity 1 patterns over finite fields, showing that sparse pattern occurrences can be eliminated with minimal recoloring.

Contribution

It introduces a new removal lemma for complexity 1 arithmetic patterns over finite fields, extending combinatorial removal concepts to an algebraic setting.

Findings

Proves an arithmetic induced removal lemma for complexity 1 patterns

Shows minimal recoloring suffices to eliminate all such patterns

Extends combinatorial removal principles to finite field arithmetic

Abstract

We prove an arithmetic analog of the induced graph removal lemma for complexity 1 patterns over finite fields. Informally speaking, we show that given a fixed collection of $r$-colored complexity 1 arithmetic patterns over $\mathbb F_q$, every coloring $\phi \colon \mathbb F_q^n \setminus\{0\} \to [r]$ with $o(1)$ density of every such pattern can be recolored on an $o(1)$-fraction of the space so that no such pattern remains.