Pointwise ergodic theorems along sequences of intermediate growth

RESUMO

We establish the first pointwise convergence result for ergodic averages with iterates along explicit and deterministic sequences of intermediate growth, that is, growing faster than any polynomial but slower than any exponential. In particular, we show that the sequence $(\lfloor \exp((\log n)^c)\rfloor)_{n\in\mathbb{Z}+}$, with $c\in(1,8/7)$, is universally $L^p$-good for every $p\in(1,\infty]$. This gives an affirmative answer to an open problem dating back to the mid 1980s and contributes to Bellow's program, initiated in the earlier part of the same decade, on the characterization of $L^p$-good sequences in pointwise ergodic theorems. The proof combines the so-called one-frequency circle method with a delicate application of Vinogradov's method for estimating exponential sums whose phases involve $\big(\lfloor \exp((\log n)^c)\rfloor\big)_{n\in\mathbb{Z}_+}$. An interesting feature of our analysis, reminiscent of estimates arising in the study of the zero-free region of the Riemann zeta function, is that the argument relies on the classical Vinogradov method, in the sense that it necessitates estimates on the number of solutions for the Vinogradov system of Diophantine equations with explicit dependence on the system's parameters.