Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Integral powers of numbers in small intervals modulo $1$: The
  cardinality gap phenomenon   This paper deals with the distribution of $\alpha \zeta^{n} \bmod 1$, where
$\alpha\neq 0,\zeta>1$ are fixed real numbers and $n$ runs through the positive
integers. Denote by $\Vert.\Vert$ the distance to the nearest integer. We
investigate the case of $\alpha\zeta^{n}$ all lying in prescribed small
intervals modulo $1$ for all large $n$, with focus on the case $\Vert\alpha
\zeta^{n}\Vert \leq \epsilon$ for small $\epsilon>0$. We are particularly
interested in what we call cardinality gap phenomena. For example for fixed
$\zeta>1$ and small $\epsilon>0$ there are at most countably many values of
$\alpha$ such that $\Vert\alpha \zeta^{n}\Vert \leq \epsilon$ for all large
$n$, whereas larger $\epsilon$ induces an uncountable set. We investigate the
value of $\epsilon$ at which the gap occurs. We will pay particular attention
to the case of algebraic and, more specific, rational $\zeta>1$. Results
concerning Pisot and Salem numbers such as some contribution to Mahler's
$3/2$-problem are implicitly deduced. We study similar questions for fixed
$\alpha\neq 0$ as well.

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The categories of this paper are: hep-ph
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
11
The categories of this paper are: hep-lat
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas