bibkey: klein2023boundedmultiplicity authors: “Jonah Klein; Dimitris Koukoulopoulos; Simon Lemieux” year: 2023 title: “On the j-th smallest modulus of a covering system with distinct moduli” doi: 10.48550/arXiv.2212.01299 url: https://arxiv.org/abs/2212.01299v2 claim: “The distortion method admits bounded numerical multiplicity; its second-moment proof supplies the square of that multiplicity as a leading factor.” strata_touched:

• D5/S3/Arith/Congruence/TwoOddPrimeUncoveredDensity license: citation-only triage: anchor

Distortion with bounded numerical multiplicity

The inspected primary version is arXiv:2212.01299v2, 23 August 2023. Page numbers below are the printed pages of that version.

Definition 2.2, p.3, defines multiplicity as the largest number of classes having the same numerical modulus. Section 3.1, pp.3–4, explicitly adapts the distortion construction to multiplicity greater than one. Lemma 3.2 and the proof of Lemma 3.3, pp.5–6, bound moments by tuples of original labels. Immediately after equation (3.2), the unsimplified bound retains a factor s^k for the kth moment of a family of multiplicity at most s. For k=2, extending the finite exponent sums gives

M_p^(2) <= s^2/(p-1)^2
  product_(q<p, q|Q)
    [1+(3q-1)/((1-delta_q)(q-1)^2)].

The parameters satisfy 0<=delta_q<=1/2. The estimate permits arbitrary finite prime-power heights and requires neither squarefree moduli nor independence of the original congruence events. The explicit Euler-factor form uses the displayed unsimplified proof; the printed conclusion of Lemma 3.3(b) is its weaker asymptotic simplification. This is an application of the published proof, not a new moment theorem or a Lean result.

Report381 already applies Theorem 3’s minimum-modulus bound for bounded multiplicity. Report348 uses the explicit second moment with the independently stated numerical continuation criterion of BBMST. Its two-cover support-intersection conclusion is a deduction combining those tools, not a theorem attributed verbatim to this source. No source text is vendored.