Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On univoque and strongly univoque sets   Much has been written about expansions of real numbers in noninteger bases.
Particularly, for a finite alphabet $\{0,1,\dots,\alpha\}$ and a real number
(base) $1<\beta<\alpha+1$, the so-called {\em univoque set} of numbers which
have a unique expansion in base $\beta$ has garnered a great deal of attention
in recent years. Motivated by recent applications of $\beta$-expansions to
Bernoulli convolutions and a certain class of self-affine functions, we
introduce the notion of a {\em strongly univoque} set. We study in detail the
set $D_\beta$ of numbers which are univoque but not strongly univoque. Our main
result is that $D_\beta$ is nonempty if and only if the number $1$ has a unique
nonterminating expansion in base $\beta$, and in that case, $D_\beta$ is
uncountable. We give a sufficient condition for $D_\beta$ to have positive
Hausdorff dimension, and show that, on the other hand, there are infinitely
many values of $\beta$ for which $D_\beta$ is uncountable but of Hausdorff
dimension zero.

16
The categories of this paper are: stat
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
7
The categories of this paper are: hep-th
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
31
The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph