Some remarks on the Balog-Wooley decomposition theorem and quantities D^+, D^\times

TL;DR

This paper investigates the properties of two set characteristics related to sum-product phenomena and improves the Balog-Wooley decomposition theorem, showing any finite real set can be partitioned into two parts with small D^+ and D^ imes.

Contribution

It introduces new variants and enhancements of the Balog-Wooley decomposition theorem, enabling the partition of finite real sets into subsets with controlled sum and product characteristics.

Findings

Any finite subset of real numbers can be split into two sets with small D^+ and D^ imes.

Provides improved bounds and variants of the Balog-Wooley decomposition theorem.

Enhances understanding of sum-product phenomena in additive combinatorics.

Abstract

In the paper we study two characteristics D^+ (A), D^\times (A) of a set A which play important role in recent results concerning sum-product phenomenon. Also we obtain several variants and improvements of the Balog-Wooley decomposition theorem. In particular, we prove that any finite subset of real numbers can be split into two sets with small quantities D^+ and D^\times.