Swaroop Hegde

Hello! I am a fifth year PhD student in the Department of Mathematics at University of Georgia, in Athens, Georgia, USA. My advisor is Giorgis Petridis.

My current research interests lie broadly in additive combinatorics, extremal combinatorics and probabilistic methods.

Prior to this, I graduated from the National Institute of Science Education and Research (NISER), Bhubaneswar, India, with an Integrated MSc degree in Mathematics.

Email: `firstname daught lastname at uga daught edu’

Publications

arXiv

An inverse and a stability result for Ruzsa's inequality on triple sumsets

Swaroop Hegde

2025

Abs arXiv

Ruzsa's inequality states that $|A+A+A| \leq |A+A|^{3/2}$ for any finite set $A$ in a commutative group. Ruzsa has constructed examples showing that this inequality is sharp asymptotically, up to a constant factor. We prove an inverse result which says that if $|A+A+A| \geq \frac{1}{M} |A+A|^{\frac{3}{2}} $ for some parameter $M,$ then the set $A$ resembles the sets in Ruzsa's construction. We then construct more families of examples which suggest that our inverse result is likely best possible qualitatively. The method extends to give an inverse result for a higher sumset analogue of Ruzsa's inequality, namely $|(h+1)A| \leq |hA|^{\frac{h+1}{h}}$ for any $h\geq 2.$ We also provide a ``99%-stability" version of Ruzsa's inequality, which describes near optimal structures when $M$ is very close to $1.$