Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Diophantine Approximations and the Convergence of Certain Series   Consider two series $$\sum_{n=1}^\infty\frac{\sin^n\pi\theta
n}{n^\alpha},\quad\sum_{n=1}^\infty\frac{\cos^n\pi\theta n}{n^\alpha}.$$ We
show that number-theoretical properties of $\theta$ have a strong effect on the
convergence when $0<\alpha\leq 1$. The complete investigation for
$\theta\in\mathbb Q$ is given. For irrational $\theta$ we prove the result
which depends on how well $\theta$ can be approximated with rational numbers,
i.e. on its irrationality measure. We obtain that if $\alpha>\frac12$ then both
series converge absolutely for almost all real $\theta$. Finally, we construct
such an everywhere dense set of $\theta$ that both series diverge when
$\alpha\leq 1$.

37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
13
The categories of this paper are: quant-ph
21
The categories of this paper are: adap-org
17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph