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Source: arXiv v2, pp. 1 and 3, Theorem 1 and its deduction from Claim 2.1 and Theorem 3.

Statement

There is an absolute constant $C>0$ such that, for every minimal covering system with distinct moduli

$$q_1<q_2<\cdots<q_k$$

and every $1\le j\le k$,

$$q_j\le\exp\left(\frac{Cj^2}{\log(j+1)}\right). \tag{1}$$

Full proof

Apply Claim 2.1 with $\ell=j$. The resulting indexed shifted tail $\mathcal C_j$ covers $\mathbb Z$, has multiplicity exactly $2^{j-1}$, and has smallest modulus $q_j$. Therefore Theorem 3 gives

$$q_j\le \exp\left( c\frac{\log^2(2^{j-1}+1)}{\log\log(2^{j-1}+2)} \right). \tag{2}$$

For $j\ge2$, $\log(2^{j-1}+1)\ll j$ and $\log\log(2^{j-1}+2)\gg\log(j+1)$, with absolute constants. Hence the exponent in (2) is $O(j^2/\log(j+1))$. Enlarging the constant handles $j=1$ and gives (1).

For $j=1$, this is an unspecified absolute minimum-modulus bound. The paper's new quantitative content is the uniform dependence on the rank $j$; it does not improve the previously published explicit bound $616000$ for the smallest modulus.

Bears on

• Problem 2, through the case $j=1$.
• Problem 1188, by constraining the ordered moduli in every minimal distinct cover, without estimating the number $F(x)$ of such systems.