---

Source: arXiv v2, p. 4, Lemma 3.1. It is the indexed-family form of Theorem 3.1 of the externally cited density paper, printed p. 386 (published PDF p. 10), whose canonical page proves that the same event-level argument permits repeated moduli.

Exact external statement

Use the finite family, events, distortion measures, and moments in the distortion setup. If

$$\sum_{j=1}^J \min\left\{M_j^{(1)}, \frac{M_j^{(2)}}{4\delta_j(1-\delta_j)}\right\}<1, \tag{1}$$

then $\mathcal A$ does not cover $\mathbb Z$.

When $\delta_j=0$, the first-moment term is used and no value is assigned to the possible $0/0$ quotient in the second term. The original density theorem also gives a quantitative lower bound for the uncovered density; the Klein--Koukoulopoulos--Lemieux argument uses only (1).

This page records an exact external input rather than reconstructing its proof. A complete treatment of the same generic removed-mass deduction appears in the square-free paper's Theorem 3.1 page, but the cited density paper remains the source of the imported result.