Sum-product inequalities with perturbation

TL;DR

This paper establishes near-optimal lower bounds on the sum of perturbed sum and product sets for a set of real numbers, extending sum-product inequalities to include controlled perturbations.

Contribution

It introduces bounds on perturbed sum and product sets using a combination of Elekes's approach and the Szemeredi-Trotter theorem, generalizing classical sum-product results.

Findings

Derived lower bounds on |P| + |S| with perturbations

Extended sum-product inequalities to perturbed sets

Utilized Szemeredi-Trotter theorem in novel context

Abstract

Suppose that A is a set of n real numbers, each at least 1 apart. Define the ``perturbed sum and product sets'' S and P to be the sums a + b + f(a,b) and products (a+g(a,b))(b+h(a,b)), where f, g, and h satisfy certain upper bounds in terms of the n, |a| and |b|. We develop almost best possible lower bounds on |P| + |S|, using the largest possible sizes of the ``perturbation parameters'' f(a,b), g(a,b) and h(a,b). Our proof uses Elekes's idea for bounding |A+A|+|A.A| from below, in combination with the Szemeredi-Trotter curve theorem (actually, a minor generalization of it) of Szekely, applied to certain polygonal arcs.