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 $\mathcal{I}$ has the Baire property if and only if there exists a positive sequence $(r_n)$ tending to zero, with $\sum_n r_n=\infty$, 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 $\mathcal{I}$ for which $o(r_n)$ is an optimal Tauberian rate for both statements (ii) and (iii).