Counting rationals and diophantine approximation in missing-digit Cantor sets

TL;DR

The paper studies how rationals distribute near missing-digit Cantor-type sets $K_{b,D}$ and analyzes intrinsic versus extrinsic Diophantine approximation in fractal settings. The authors introduce the Fourier $\ell^1$-dimension $\hat{\kappa}_1(\nu)$ of missing-digit measures and prove that, under AD-regularity and $\hat{\kappa}_1(\nu)>1/2$, one obtains quantitative bounds on the counting function $\mathcal{N}_K(T)$ and zero-measure results for intrinsic approximation sets $W_K(\alpha)$ near the critical exponent. They reduce the main theorems to estimating $\hat{\kappa}_1(\nu)$, develop a concrete algorithm to approximate this dimension via $S_L$-based bounds, and provide analytic and numerical bounds to certify the dimension for large bases and to compute it for smaller bases. The results advance conjectures of Broderick–Fishman–Reich and extend BD-type and Levesley–Salm–Velani–type conjectures to a broad class of missing-digit fractals, offering new tools to quantify rational points and Diophantine properties on self-similar sets.

Abstract

We establish a new upper bound for the number of rationals up to a given height in a missing-digit set, making progress towards a conjecture of Broderick, Fishman, and Reich. This enables us to make novel progress towards another conjecture of those authors about the corresponding intrinsic diophantine approximation problem. Moreover, we make further progress towards conjectures of Bugeaud--Durand and Levesley--Salp--Velani on the distribution of diophantine exponents in missing-digit sets. A key tool in our study is Fourier $\ell^1$ dimension introduced by the last named author in [H. Yu, Rational points near self-similar sets, arXiv:2101.05910]. An important technical contribution of the paper is a method to compute this quantity.

Theorems & Definitions (49)

• Conjecture 1.1
• Theorem 1.2: Main Theorem 1
• Conjecture 1.3: Broderick, Fishman and Reich BFR2011
• Theorem 1.4: Main Theorem 2
• Remark 1.5
• Conjecture 1.6
• Conjecture 1.7
• Theorem 1.8: Main Theorem 3
• Theorem 2.1
• Remark 2.2