Erdős Problem 42: A Fourier--compactness argument

This repository contains a self-contained proof of Erdős Problem 42 using a Fourier--compactness argument. The argument reduces the problem to a one-sided Fourier lemma for Cayley graphs over prime cyclic finite fields.

Theorem (Erdős Problem 42)

For every $M \ge 1$ there is $N_0(M)$ such that, whenever $N \ge N_0(M)$ and $A \subset {1, \ldots, N}$ is a Sidon set, there is a Sidon set $B \subset {1, \ldots, N}$ with $|B| = M$ and $(A - A) \cap (B - B) = {0}$.

Repository Contents

• erdos42_XL_en_referenced.tex: LaTeX source for the English version of the paper.
• erdos42_XL_en_referenced.pdf: Compiled English version of the paper.
• erdos42_XL_it_referenced.tex: LaTeX source for the Italian version of the paper.
• erdos42_XL_it_referenced.pdf: Compiled Italian version of the paper.

Key Technical Points

The proof utilizes:

• A one-sided Cayley--Fourier lemma.
• A multicolour weighted degree-one compactness/counting proposition.
• Ultraproduct Fourier factors and degree-one generalized von Neumann estimates.
• Spectral passage from finite one-sided Fourier lower bounds to Fourier positivity on the compact limit.

Acknowledgments

The public discussion attached to the Erdős Problems page contains relevant partial progress, reductions, and comments on the proposed solution. Specifically, contributions from Zach Hunter, Daniil Sedov, Harjas, qrdl, Thomas F. Bloom, Will Sawin, and Terence Tao are noted and cited within the paper.