A non-linear Roth theorem for thick Cantor sets

2025/09/22 by McDonald, Alex, Nguyen, Micah
#28A80 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.17880

Abstract

We prove that for any function f satisfying certain mild conditions and any Cantor set K with Newhouse thickness greater than 1, there exists x∈ K and t>0 such that \x-t,x,x+f(t)\⊂ K. This is an extension of previous work on the existence of three-term arithmetic progressions in Cantor sets to the non-linear setting.

Citations

• On Polynomial Progressions Inside Sets of Large Dimension
• New improvement to Falconer distance set problem in higher dimensions
• A survey on Newhouse thickness, fractal intersections and patterns
• Finite Point configurations in Products of Thick Cantor sets and a Robust Nonlinear Newhouse Gap Lemma
• Patterns in thick compact sets
• On Falconer's distance set problem in the plane
• Interior of sums of planar sets and curves
• On Certain Sets of Integers

Related

• Descriptions of Cantor Sets: A Set‑Theoretic Survey and Open Problems
• Four-term progression free sets with three-term progressions in all large subsets
• Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions
• Multiplication on uniform λ-Cantor sets
• A nonlinear version of the Newhouse thickness theorem