Timeline for Can all partial sums $\sum_{k=1}^n f(ka)$ where $f(x)=\log|2\sin(x/2)|$ be non-negative?
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10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 29, 2021 at 14:31 | vote | accept | fedja | ||
| Nov 29, 2021 at 14:16 | answer | added | Aleksei Kulikov | timeline score: 7 | |
| Oct 18, 2021 at 19:38 | comment | added | Anthony Quas | So I looked at the 1987 paper of Peres. I hadn't realized the distinction between what he is doing and what you are asking. He shows that for a fixed cts dynam sys $T$ on a compact space, a fixed cts fn $g$ and a fixed invariant measure, there is some point $x$ such that $\sum_{j=0}^{n-1}g(T^jx)\ge n\int g\,d\mu$ for all $n\in\mathbb N$. The lemma in that paper also works for your function $g$ taking values in $[-\infty,\infty)$ so that for any $a$, there exists an $x$ such that $g(x+a)+\ldots+g(x+na)\ge n\int g$ for all $n$. Your question, though was about $x=0$. Peres does not answer that. | |
| Oct 15, 2021 at 16:13 | comment | added | juan | Today appeared in arXiv a paper arXiv:2110.07407v1 "A conjecture of Zagier and the value distribution of quantum modular forms" by Aistleitner and Borda, that have related material and references in particular the paper cited by @Kulikov and others by the same authors. | |
| Oct 15, 2021 at 11:57 | comment | added | Mateusz Kwaśnicki | Just a remark: judging by the outcome of numerical experiments, the choice $a = 2\pi\sqrt2$ seems to maximize the infimum of the sequence of sums (as you certainly know, but it might be interesting for the others). | |
| Oct 15, 2021 at 6:40 | comment | added | Aleksei Kulikov | I don't know about non-negativity, but this beautiful paper shows that there is a uniform lower bound for this sum in the case $a = \pi(\sqrt{5}-1)$. There's also a lot of references, in particular about the continued fraction expansion of the potential $a$. | |
| Oct 15, 2021 at 5:00 | comment | added | Anthony Quas | Yuval Peres and David Ralston have some papers on a closely related topic. They deal with continuous functions (unlike your function with a singularity), and prove the existence of what they call heavy points: these are exactly the points you are looking for where all sums are non-negative. | |
| Oct 15, 2021 at 4:08 | comment | added | fedja | @mathworker21 No. I agree that it has the same spirit but I do not immediately see whether the techniques there yield anything in my case :-) | |
| Oct 15, 2021 at 0:50 | comment | added | mathworker21 | have u seen B6 and solutions? | |
| Oct 15, 2021 at 0:20 | history | asked | fedja | CC BY-SA 4.0 |