Sets with distinct sums of pairs, long arithmetic progressions, and continuous mappings

TL;DR

The paper proves that continuous functions with a nonzero measure of nonlinearity can map sets containing arbitrarily long arithmetic progressions onto sets with distinct sums of pairs, revealing new connections between structure and nonlinearity.

Contribution

It establishes a novel link between the nonlinearity measure of continuous functions and their ability to map structured sets onto sets with distinct pairwise sums.

Findings

Continuous functions with nonzero nonlinearity measure can map sets with long arithmetic progressions onto sets with distinct sums of pairs.

The result connects the concepts of arithmetic progressions, nonlinearity, and sumsets in a new way.

The paper provides conditions under which such mappings exist, expanding understanding of structure-preserving transformations.

Abstract

We show that if $\varphi \colon \mathbb R\rightarrow\mathbb R$ is a continuous mapping and the set of nonlinearity of $\varphi$ has nonzero Lebesgue measure, then $\varphi$ maps bijectively a certain set that contains arbitrarily long arithmetic progressions onto a certain set with distinct sums of pairs.