Problem 27

An $\epsilon$ -almost covering system is a set of congruences $a_i\pmod{n_i}$ for distinct moduli $n_1<\cdots<n_k$ such that the density of those integers which satisfy none of them is $\leq \epsilon$ .

Is there a constant $C>1$ such that for every $\epsilon>0$ and $N\geq 1$ there is an $\epsilon$ -almost covering system with $N\leq n_1<\cdots <n_k\leq CN$ ?

Source: erdosproblems.com/27

Number theory
Covering systems

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SolvedDisproved

An accepted solution exists. The statement is false.

Disproved, the site's label. The answer is no, by Filaseta, Ford, Konyagin, Pomerance and Yu (J. Amer. Math. Soc. 2007, refereed), whom the site credits; the acceptance evidence is on their claim page (Filaseta, Ford, Konyagin, Pomerance and Yu, 2005). A second proof, Theorem 5.1 of Balister, Bollobás, Morris, Sahasrabudhe and Tiba (Invent. Math. 2022, refereed), has its own claim page (Balister–Bollobás–Morris–Sahasrabudhe–Tiba, 2018).

BBMST22: Balister, Paul and Bollobás, Béla and Morris, Robert and Sahasrabudhe, Julian and Tiba, Marius, On the Erdős covering problem: the density of the uncovered set. Invent. Math. (2022), 377-414.
FFKPY07: Filaseta, Michael and Ford, Kevin and Konyagin, Sergei and Pomerance, Carl and Yu, Gang, Sieving by large integers and covering systems of congruences. J. Amer. Math. Soc. (2007), 495-517.
Ho15: Hough, Bob, Solution of the minimum modulus problem for covering systems. Ann. of Math. (2) (2015), 361-382.

Problem 27

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