Perfect fractal sets with zero Fourier dimension and arbitrarily long arithmetic progressions

TL;DR

This paper constructs perfect Moran fractal sets with any Hausdorff dimension between 0 and 1, zero Fourier dimension, and arbitrarily long arithmetic progressions, revealing complex structure in fractals.

Contribution

It introduces a method to create fractals with prescribed Hausdorff dimension, zero Fourier dimension, and arbitrarily long arithmetic progressions, advancing understanding of fractal harmonic analysis.

Findings

Existence of perfect fractals with any Hausdorff dimension and zero Fourier dimension.

Construction of fractals containing arbitrarily long arithmetic progressions.

Demonstration of complex harmonic properties in fractals with prescribed dimensions.

Abstract

By considering a Moran-type construction of fractals on $[0,1]$, we show that for any $0\le s\le 1$, there exists some Moran fractal set, which is perfect, with Hausdorff dimension $s$ whose Fourier dimension is zero and it contains arbitrarily long arithmetic progressions.