Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the distribution of powers of real numbers modulo 1   Given a strictly increasing sequence of positive real numbers tending to
infinity $(q_{n})_{n=1}^{\infty}$, and an arbitrary sequence of real numbers
$(r_{n})_{n=1}^{\infty}.$ We study the set of $\alpha\in(1,\infty)$ for which
$\lim_{n\to\infty}\|\alpha^{q_{n}}-r_{n}\|= 0$. In \cite{Dub} Dubickas showed
that whenever $\lim_{n\to\infty}(q_{n+1}-q_{n})=\infty,$ there always exists a
transcendental $\alpha$ for which $\lim_{n\to\infty}\|\alpha^{q_{n}}-r_{n}\|=
0.$ Adapting the approach of Bugeaud and Moshchevitin \cite{BugMos}, we improve
upon this result and show that whenever
$\lim_{n\to\infty}(q_{n+1}-q_{n})=\infty,$ the set of $\alpha\in(1,\infty)$
satisfying $\lim_{n\to\infty}\|\alpha^{q_{n}}-r_{n}\|= 0$ is a dense set of
Hausdorff dimension $1$.

21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph