On Assouad dimension and arithmetic progressions in sets defined by digit restrictions

2017/10/19 by Jinjun Li, Min Wu, Li, Jinjun +3 · 1 citation
Mathematics · #28A80 #42A38 #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1710.07144

openalex publication_date 2017/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the set defined by digit restrictions contains arbitrarily long arithmetic progressions if and only if its Assouad dimension is one. Moreover, we show that for any 0≤ s≤ 1, there exists some set on ℝ with Hausdorff dimension s whose Fourier dimension is zero and it contains arbitrarily long arithmetic progressions.

Citations

• On Some Sequences of Integers
• Fractal Geometry: Mathematical Foundations and Applications.
• On Certain Sets of Integers

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• The generalized upper box dimension

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