Tauberian rates and the Baire property of ideals
Abstract
We study Tauberian conditions on the successive differences of real sequences which ensure that: (i) ordinary convergence coincides with ideal convergence; or (ii) every ordinary accumulation point is an ideal cluster point; or (iii) every ordinary accumulation point is an ideal limit point. First, we prove that an ideal ℐ has the Baire property if and only if there exists a positive sequence (rn) tending to zero, with ∑n rn=∞, such that |xn+1−xn|=O(rn) is a Tauberian condition for any of the three previous statements. Second, for analytic P-ideals, we characterize the validity of the Tauberian rate O(1/n) in terms of properties of lower semicontinuous submeasures which represent it. Lastly, in the case of statistical limit points, we prove that o(1/n) is a Tauberian condition for the statement (iii), whereas O(1/n) need not suffice: in this sense, o(1/n) is an optimal Tauberian rate for (iii). On the other hand, for statistical cluster points in (ii), O(1/n) is an optimal one among nonincreasing rates. For every positive null sequence (rn) with divergent sum, we also construct a summable ideal ℐ for which o(rn) is an optimal Tauberian rate for both statements (ii) and (iii).
Related
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- A baby Baire property
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- On the Multiplicative Thom-Sebastiani Property for Bernstein-Sato Polynomials of Ideals
BibTeX
@misc{leonetti2026tauberian,
title = {{Tauberian rates and the Baire property of ideals}},
author = {Paolo Leonetti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21889},
eprint = {2610.21889},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21889},
}
RIS
TY - UNPB
TI - Tauberian rates and the Baire property of ideals
AU - Paolo Leonetti
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21889
UR - https://arxiv.org/abs/2610.21889
AB - We study Tauberian conditions on the successive differences of real sequences which ensure that: (i) ordinary convergence coincides with ideal convergence; or (ii) every ordinary accumulation point is an ideal cluster point; or (iii) every ordinary accumulation point is an ideal limit point. First, we prove that an ideal ℐ has the Baire property if and only if there exists a positive sequence (r_n) tending to zero, with ∑_n r_n=∞, such that |x_n+1−x_n|=O(r_n) is a Tauberian condition for any of the three previous statements. Second, for analytic P-ideals, we characterize the validity of the Tauberian rate O(1/n) in terms of properties of lower semicontinuous submeasures which represent it. Lastly, in the case of statistical limit points, we prove that o(1/n) is a Tauberian condition for the statement (iii), whereas O(1/n) need not suffice: in this sense, o(1/n) is an optimal Tauberian rate for (iii). On the other hand, for statistical cluster points in (ii), O(1/n) is an optimal one among nonincreasing rates. For every positive null sequence (r_n) with divergent sum, we also construct a summable ideal ℐ for which o(r_n) is an optimal Tauberian rate for both statements (ii) and (iii).
ER -