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Source claim

For any finite set or multiset $M$ of moduli, v2 Lemma 4.10 claims that the density covered by any corresponding residue system is at most

$$\sum_{\substack{\varnothing\ne S\subseteq M\\ S\text{ pairwise coprime}}} \frac{(-1)^{|S|+1}}{\operatorname{lcm}(S)}.$$

For a multiset, the occurrences are counted separately. This is the interpretation explicitly used on pp. 11–12, where the source weights a subset of size $k$ by the multiplicity raised to the $k$th power.

The unrestricted multiset claim is false. This page does not assert a counterexample to every distinct-modulus variant or to the ordinary CRT formula for pairwise coprime moduli.

Complete counterexample and convention

Take three indexed occurrences of modulus 2 and four of modulus 3, with classes

$$0,1,2\pmod2,\qquad 0,1,2,3\pmod3.$$

The first two classes already cover all integers, so the covered density is one. The sum of singleton contributions is $3/2+4/3$. There are $3\cdot4=12$ pairwise coprime pairs, each consisting of one occurrence of 2 and one of 3, and each contributing $-1/6$. No triple is pairwise coprime. Thus the asserted upper bound is

$$\frac32+\frac43-\frac{12}{6}=\frac56<1.$$

This indexed system intentionally repeats congruence classes: for example, $0\pmod2$ and $2\pmod2$ are the same subset of the integers. Such repetition is allowed by the stated arbitrary-multiset claim and by treating all indexed occurrences in its sum. If an additional convention forbids repeated classes, this auxiliary example would fall outside that narrower domain; that restriction would have to be stated and checked in every application.

The distinct-divisor covering used in the separate counterexample to Theorem 4.11 has no repeated modulus or class. That counterexample directly satisfies the printed theorem's hypotheses and does not depend on this convention.

Source and scope

Canonical arXiv v2, p. 10, Lemma 4.10; the multiplicity interpretation is explicit in the proof of Theorem 4.11 on pp. 11–12. This is a compilation-supplied source correction, not an author-issued erratum. The full journal version and the numerical density computations have not been checked here.

Bears on

• Problem 7: qualifications on a proposed residue-coverage bound and on its use with repeated moduli.