Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Erd\H{o}s Covering Problem: the density of the uncovered set   Since their introduction by Erd\H{o}s in 1950, covering systems (that is,
finite collections of arithmetic progressions that cover the integers) have
been extensively studied, and numerous questions and conjectures have been
posed regarding the existence of covering systems with various properties. In
particular, Erd\H{o}s asked if the moduli can be distinct and all arbitrarily
large, Erd\H{o}s and Selfridge asked if the moduli can be distinct and all odd,
and Schinzel conjectured that in any covering system there exists a pair of
moduli, one of which divides the other.
  Another beautiful conjecture, proposed by Erd\H{o}s and Graham in 1980,
states that if the moduli are distinct elements of the interval $[n,Cn]$, and
$n$ is sufficiently large, then the density of integers uncovered by the union
is bounded below by a constant (depending only on $C$). This conjecture was
confirmed (in a strong form) by Filaseta, Ford, Konyagin, Pomerance and Yu in
2007, who moreover asked whether the same conclusion holds if the moduli are
distinct and sufficiently large, and $\sum_{i=1}^k \frac{1}{d_i} < C$. Although
this condition turns out not to be sufficiently strong to imply the desired
conclusion, as the main result of this paper we will give an essentially best
possible condition which is sufficient.
  Our method has a number of further applications. Most importantly, we prove
the conjecture of Schinzel stated above, which was made in 1967. We moreover
give an alternative (somewhat simpler) proof of a breakthrough result of Hough,
who resolved Erd\H{o}s' minimum modulus problem, with an improved bound on the
smallest difference. Finally, we make further progress on the problem of
Erd\H{o}s and Selfridge.

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The categories of this paper are: hep-lat
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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: eess
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The categories of this paper are: astro-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: math-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph
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The categories of this paper are: econ
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-th