Improving cardinality estimation of sums of sets with convexity

TL;DR

This paper introduces a refined method for estimating the size of sumsets of convex sets with specific curvature properties, leveraging advanced incidence geometry and spectral techniques to improve previous bounds.

Contribution

It provides a novel improvement in sumset cardinality estimation for convex sets with negative discrete third derivative, building on incidence geometry and spectral methods.

Findings

Enhanced estimate accuracy for sumsets of convex sets

Application of incidence geometry techniques to sumset problems

Utilization of spectral methods for improved bounds

Abstract

This paper presents a slight improvement of the estimate of sumsets of convex sets with negative discrete third derivative. The proposed method is based on some previous works in incidence geometry and use of spectrum method developed earlier for notable progresses in this fields.