Sumset Lower Bounds In Abelian Groups

Author

• Van T. Huynh, University of MississippiFollow

Date of Award

Spring 5-1-2026

Document Type

Undergraduate Thesis

Department

Mathematics

First Advisor

Thai Hoang Le

Second Advisor

Micah Milinovich

Third Advisor

Josh Hendrickson

Relational Format

Dissertation/Thesis

Abstract

This thesis investigates sumset lower bounds across discrete and continuous settings. We begin with general inequalities in torsion-free abelian groups and then specialize to the integers modulo prime p, where we present the Cauchy–Davenport Theorem, which establishes the bound ∣A+B∣≥min(p,∣A∣+∣B∣−1). The equality case is further examined via Vosper's Theorem, which characterizes subsets attaining this bound as arithmetic progressions under suitable conditions. The continuous analogue in Euclidean spaces is then considered, where cardinality is replaced by Lebesgue measure. In this setting, the Brunn–Minkowski Inequality provides a sharp lower bound for the Lebesgue measure of A+B and serves as a geometric counterpart to discrete sumset estimates. As an application, we discuss its connection to the isoperimetric inequality, illustrating the interplay between additive and geometric structures.

Recommended Citation

Huynh, Van T., "Sumset Lower Bounds In Abelian Groups" (2026). Honors Theses. 3512.
https://egrove.olemiss.edu/hon_thesis/3512