Weighted density bounds for covering systems over Dedekind domains

Abstract

We study the density of uncovered sets arising from finite families of residue classes over Dedekind domains with finite residue norms. Building on the distortion method of Balister, Bollobás, Morris, Sahasrabudhe and Tiba, we prove a general weighted density criterion in which the weight attached to a prime ideal is allowed to vary. In the critical Mertens regime this yields, for every \(\varepsilon>0\), the admissible prime-ideal weight \[ 1+\frac{(\log N\mathfrak p)^3(\log\log N\mathfrak p)^{1+\varepsilon}}{N\mathfrak p}, \] up to a harmless modification at finitely many small prime ideals. This is asymptotically smaller than the fixed weight \(1+(\log N\mathfrak p)^{3+\varepsilon}/N\mathfrak p\) appearing in the earlier density theorem. The result is valid for systems of any fixed multiplicity and, for multiplicity one, requires the ideal moduli themselves to be distinct but not their norms. We then place the analytic hypotheses in Knopfmacher's Axiom~A framework. In the critical case of Axiom~A with exponent \(1\), the prime-ideal estimates needed by the weighted theorem are automatic. More generally, under Axiom~A with exponent \(ρ>0\), we prove a weighted mean-value theorem on ideals; in the subcritical range \(0<ρ<1\), a first-moment argument shows that sufficiently large moduli cover an arbitrarily small proportion of the ambient space. We also explain that the distortion construction itself only uses a compatible system of finite residue spaces with uniform fibres and a Chinese-remainder-type product structure, suggesting a possible extension beyond Dedekind domains. Finally, we show that a direct passage from the second moment to higher integer moments does not lower the logarithmic exponent \(3\) within this moment hierarchy.