December, 2020

93-98

Vsevolod F. Lev

Let p ≥ 3 be a prime, S⊆𝔽p2 a nonempty set, and w:𝔽p2→R a function with supp w = S. Applying an uncertainty inequality due to András Bíró and the present author, we show that there are at most 12|S| directions in 𝔽p2 such that for every line l in any of these directions, one has ∑z∈lw(z)=1p∑z∈𝔽p2w(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 12|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.

Lev, V.F.. (2020). Point Distribution and Perfect Directions in 𝔽p2\mathbb{F}_p^2. Uniform distribution theory, 15(2), 93-98. https://doi.org/10.2478/udt-2020-0012