2024/05/15 by A. Shapira, Shapira, Asaf
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Benford’s Law and Fraud Detection #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2405.09402
openalex publication_date 2024/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A linear equation E is said to be sparse if there is c>0 so that every subset of [n] of size n1-c contains a solution of E in distinct integers. The problem of characterizing the sparse equations, first raised by Ruzsa in the 90's, is one of the most important open problems in additive combinatorics. We say that E in k variables is abundant if every subset of [n] of size ε n contains at least poly(ε)⋅ nk-1 solutions of E. It is clear that every abundant E is sparse, and Girão, Hurley, Illingworth and Michel asked if the converse implication also holds. In this note we show that this is the case for every E in 4 variables. We further discuss a generalization of this problem which applies to all linear equations.