On binomial sums, additive energies, and lazy random walks

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.