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

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.

Latest version and referee comments incorporated

References in corpus (2)

• Arithmetic progressions in sets of fractional dimension
• Long progressions in sets of fractional dimension