On binomial sums, additive energies, and lazy random walks

TL;DR

This paper proves a sharp estimate for additive energies in the discrete hypercube, generalizing previous results, and provides an elementary inequality with a probabilistic interpretation related to lazy random walks.

Contribution

It establishes a key inequality for additive energies, completing a conjecture and extending prior work with a novel elementary approach and probabilistic insight.

Findings

Proved a sharp estimate for k-additive energies of hypercube subsets.

Provided an elementary inequality previously verified only for k ≤ 100.

Connected the inequality to lazy non-symmetric random walks on integers.

Abstract

We establish a sharp estimate for $k$-additive energies of subsets of the discrete hypercube conjectured by de Dios Pont, Greenfeld, Ivanisvili, and Madrid in arXiv:2112.09352, which generalizes a result by Kane and Tao. This note proves the only missing ingredient, which is an elementary inequality for real numbers, previously verified only for $k\leq100$. We also give an interpretation of this inequality in terms of a lazy non-symmetric simple random walk on the integer lattice.