The box dimension of a (finitely generated) self-similar set always exists. However, perhaps surprisingly, this is not the case in the infinitely generated case. If we only consider the lower and upper box dimensions in isolation, the situation seems mysterious. But if we take into account information about all scales simultaneously, a cohesive picture emerges. One way to understand this phenomenon is to use a certain Lipschitz function called the branching function which one can associate with general subsets of Euclidean space. This function is ubiquitous in (continuous) incidence geometry but its appearance in the context of dynamically invariant sets is quite surprising. Based on joint work with A. Banaji.
October 2025
Monday, October 20, 2025
10:15-10:55 am, room 306 in Building H, BME.
Topology of univoque sets in double-base expansions
This is a joint work with Yuru Zou and Yichang Li. Given two real numbers \(q_0, q_1>1\) satisfying \(q_0+q_1
\geq q_0 q_1\) and two real numbers \(d_0 \neq d_1\), by a double-base expansion of a real number \(x\) we
mean a sequence \(\left(i_k\right) \in\{0,1\}^{\infty}\) such that
$$
x=\sum_{k=1}^{\infty} \frac{d_{i_k}}{q_{i_1} q_{i_2} \cdots q_{i_k}} .
$$
We denote by \(U_{q_0, q_1}\) the set of numbers \(x\) having a unique expansion.
Its topological properties have been investigated in the equal-base case \(q_0=q_1\) for a long time.
We extend this research to the case \(q_0 \neq q_1\).
While many results remain valid, a great number of new phenomena appear due to the increased complexity of
double-base expansions.