Year: 2025
Type: article
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 logp factor), the bound recovers the optimal growth pn−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.
Source: Mathematics
DOI: 10.3390/math13213374
Author: Francisco-Javier Soto
Institution: Universidad Rey Juan Carlos
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,137
OpenAlex ID: W4415463913