Intersecting well approximable and missing digit sets
Bing Li, Sanju Velani, Bo Wang
TL;DR
This work analyzes the size and dimension of the intersection between ψ-well-approximable numbers $W_t(\psi)$ and missing-digit self-similar sets $C(b,D)$. It establishes a general bound $\dim_{\rm H}(W_t(\psi)\cap C(b,D)) \le \dim_{\rm H}(W_t(\psi))\cdot\dim_{\rm H}(C(b,D))$ and, under additional restrictions when $b$ and $t$ share prime divisors, a zero-measure criterion and a sharp dimension formula that refines prior results. Crucially, it shows that the product formula for the intersection’s dimension can fail when $b$ and $t$ are multiplicatively independent, providing explicit constructions that disprove the LLW product conjecture. A key corollary extends the dimension bound to the intersection with any self-similar set, highlighting a universal upper bound that improves earlier estimates. The results close gaps in the LLW framework and illuminate the intricate interaction between Diophantine approximation and fractal geometry.
Abstract
Let $b\geq3$ be an integer and $C(b,D)$ be the set of real numbers in $[0,1]$ whose $b$-ary expansion consists of digits restricted to a given set $D\subseteq\{0,\ldots,b-1\}$. Given an integer $t\geq2$ and a real, positive function $ψ$, let $W_{t}(ψ)$ denote the set of $x$ in $[0,1]$ for which $|x-p/t^{n}|<ψ(n)$ for infinitely many $(p,n)\in\mathbb{Z}\times\mathbb{N}$. We prove a general Hausdorff dimension result concerning the intersection of $W_{t}(ψ)$ with an arbitrary self similar set which implies that $\dim_{\rm H}(W_{t}(ψ)\cap C(b,D))\le\dim_{\rm H}W_{t}(ψ)\times \dim_{\rm H}C(b,D)$. When $b$ and $t$ have the same prime divisors, under certain restrictions on the digit set $D$, we give a sufficient condition for the Hausdorff measure of $W_{t}(ψ)\cap C(b,D)$ to be zero. This closes a gap in a result of Li, Li and Wu \cite{LLW2025} and shows that the dimension of the intersection can be strictly less than the product of the dimensions. The latter disproves the product conjecture of Li, Li and Wu.