Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Two Erdos problems on lacunary sequences: Chromatic number and
  Diophantine approximation   Let ${n_k}$ be an increasing lacunary sequence, i.e., $n_{k+1}/n_k>1+r$ for
some $r>0$. In 1987, P. Erdos asked for the chromatic number of a graph $G$ on
the integers, where two integers $a,b$ are connected by an edge iff their
difference $|a-b|$ is in the sequence ${n_k}$. Y. Katznelson found a connection
to a Diophantine approximation problem (also due to Erdos): the existence of
$x$ in $(0,1)$ such that all the multiples $n_j x$ are at least distance
$\delta(x)>0$ from the set of integers. Katznelson bounded the chromatic number
of $G$ by $Cr^{-2}|\log r|$. We apply the Lov\'asz local lemma to establish
that $\delta(x)>cr|\log r|^{-1}$ for some $x$, which implies that the chromatic
number of $G$ is at most $Cr^{-1} |\log r|$. This is sharp up to the
logarithmic factor.

36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
17
The categories of this paper are: q-fin
33
The categories of this paper are: plasm-ph
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int