Counting Rainbow Solutions of a Linear Equation over F p via Fourier-Analytic Methods

Francisco-Javier Soto ()
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Francisco-Javier Soto: Department of Computer Science and Statistics, Escuela Técnica Superior de Ingeniería Informática, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain

Mathematics, 2025, vol. 13, issue 21, 1-9

Abstract: We study rainbow solutions to linear equations modulo a prime p , where the residue classes are partitioned into n color classes. Using the Fourier method, we derive a universal lower bound that depends only on the class densities and a single spectral parameter: the Fourier bias (the largest nontrivial Fourier coefficient) of each class. When the biases are at the square-root cancellation scale p − 1 / 2 (for random colorings, up to a log p factor), the bound recovers the optimal growth p n − 1 with an explicit leading constant and negligible error. Our results complement recent work: in low-bias regimes (pseudorandom or random) they yield sharper quantitative bounds with transparent constants, and the bound requires no extra hypotheses such as coefficient separability.

Keywords: rainbow solutions; finite fields; additive combinatorics; Fourier analysis; Fourier bias; Gauss sums; random colorings (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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