Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 1009-1017

PERFECT FRACTAL SETS WITH ZERO FOURIER DIMENSION AND ARBITRARY LONG ARITHMETIC PROGRESSION

Chun-Kit Lai

San Francisco State University, Department of Mathematic
1600 Holloway Avenue, San Francisco, CA 94132, U.S.A.; cklai 'at' sfsu.edu

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

2010 Mathematics Subject Classification: Primary 28A78, 42A38.

Key words: Arithmetic progression, Fourier dimension, fractals, perfect sets, Moran sets.

Reference to this article: C.-K. Lai: Perfect fractal sets with zero Fourier dimension and arbitrarily long arithmetic progressions. Ann. Acad. Sci. Fenn. Math. 42 (2017), 1009-1017.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4263