2019/03/04 by Vsevolod F. Lev, Lev, Vsevolod F.
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1903.01518
openalex publication_date 2019/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p≥ 3 be a prime, S⊆\mathbb Fp2 a nonempty set, and w\colon\mathbb Fp2→\mathbb R a function with supp w=S. Applying an uncertainty inequality due to András Biró and the present author, we show that there are at most \frac12|S| directions in \mathbb Fp2 such that for every line l in any of these directions, one has ∑z∈ l w(z) = \frac1p∑z∈\mathbb Fp2 w(z), except if S itself is a line and w is constant on S (in which case all, but one direction have the property in question). The bound \frac12|S| is sharp. As an application, we give a new proof of a result of Rédei-Megyesi about the number of directions determined by a set in a finite affine plane.