A proof of Fraenkel’s conjecture

Abstract: Fraenkel’s conjecture asserts that any partition of the integers into $m \geq 3$ Beatty sequences with pairwise distinct moduli must have densities $2^j/(2^m - 1)$ for $j = 0, …, m - 1$. In this talk, I will present a proof whose main ingredient is a density bound independent of the number of parts: every such partition contains a sequence of density at least 1/3. Building on the discrete Fourier framework developed by Graham and O’Bryant, we establish this bound by combining analytic estimates with exact finite verification using integer and rational arithmetic. The bound supplies the missing input for Tijdeman’s deletion method, allowing us to recover the binary density pattern by induction. I will explain how the Fourier argument connects with the combinatorial reconstruction and discuss the role of the finite computations. This is joint work with Professor Ying Zhang.

commutative algebracombinatoricsnumber theory

Audience: researchers in the topic

New York Number Theory Seminar

Series comments: Meeting ID: 840 6618 4717 Passcode: 304403