Existence of auxiliary regularly varying functions g* in boundary-index cases

Ascertain whether, for any regularly varying comparison function g with index 0 or 2 that satisfies the hypotheses in the sparse-spectrum criteria, there exists a regularly varying function g* whose reciprocal is locally integrable and which fulfills the required asymptotic identities: for Ind g = 0, ∫_t^∞ ∫_u^∞ (1/g*(s)) (ds/s) (du/u) \asymp 1/g(t); for Ind g = 2, ∫_1^t (s^2/g*(s)) (ds/s) \asymp t^2/g(t).

Background

The sparse-spectrum results (convergence class and type via the Weyl coefficient approach) require, in boundary-index cases Ind g ∈ {0,2}, the existence of an auxiliary regularly varying function g* that links certain integral transforms to 1/g or t2/g, respectively. The existence of such a g* enables pointwise characterizations of summability of the spectrum or type conditions.

While explicit constructions of g* are available for specific Lindelöf-type g, a general existence theorem would significantly broaden the applicability of the sparse-spectrum criteria and clarify the structure of boundary-index regimes.

References

In the boundary cases Ind g\in{0,2} we do not known whether a function g* with the required properties can always be found (see the remarks below for a class of functions g where g* can be found).

— Spectral properties of canonical systems: discreteness and distribution of eigenvalues  (2504.00182 - Reiffenstein et al., 31 Mar 2025) in Following Theorem U56, Section “Trace class and sparse spectrum: the Weyl coefficient approach” (U110)

Which conditions on g, weaker than the weighted mass condition eq:xi_mass, still force eq:xi_reciprocal at a transparent exponent?

eq:xi_mass:

Vβ=∫(0,1)t−β d∣ν∣(t)<∞.\mathcal V_\beta=\int_{(0,1)}t^{-\beta}\,d|\nu|(t)<\infty .

eq:xi_reciprocal:

Ξg(β)=1g∗(β),\Xi_g(\beta)=\frac{1}{g^{*}(\beta)},

— Regular Arithmetic Functions, Volume I. Theory, Applications, Examples  (2609.09366 - Cloitre, 8 Sep 2026) in Open Problem 2.6, immediately after Proposition 2.5