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Let $f(x)=\log|2\sin(x/2)|$ (the normalizing factor $2$ is chosen to have the average over the period equal to $0$). Does there exist $a>0$ such that all sums $\sum_{k=1}^n f(ak)\ge 0$? The computations (run up to the values of $n$ where I could not rely on the floating point precision any more) show that it may be the case even for $a=2\pi\sqrt 2$ but I'll be happy with any $a$ (or with a proof that no such $a$ exists).

This question came up in our joint study of Leja interpolation points with Volodymyr Andriyevskyy but the connection is a bit too convoluted to be explained here :-)

fedja
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    have u seen B6 and solutions? – mathworker21 Oct 15 '21 at 00:50
  • @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 :-) – fedja Oct 15 '21 at 04:08
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    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. – Anthony Quas Oct 15 '21 at 05:00
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    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 $a$. – Aleksei Kulikov Oct 15 '21 at 06:40
  • 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). – Mateusz Kwaśnicki Oct 15 '21 at 11:57
  • 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. – juan Oct 15 '21 at 16:13
  • 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. – Anthony Quas Oct 18 '21 at 19:38

1 Answers1

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The answer to this question is yes, which is proved in the following paper that appeared on arXiv today. Specifically, part $(ii)$ of Theorem 2 of this paper, in the case $b = 1$, says that for $\beta = \frac{\sqrt{5}-1}{2}$ the smallest term of the sequence $$P_N(\beta) = \prod\limits_{r = 1}^N 2|\sin(\pi r\beta)|$$ is the first term $P_1(\beta)$ (this is the same as the sequence in OP up to denoting $a = 2\pi \beta$ and taking the exponent). Since $$P_1(\beta) = 2|\sin(\pi \beta)| = 1,86\ldots > 1,$$ the sequence in the OP for $a = 2\pi \beta$ is strictly positive and even has a uniform poisitve lower bound $\log(1,86\ldots) = 0,62\ldots$ .

Note also that the case $b = 2$ of the same Theorem gives us $\beta = \sqrt{2}-1$. By periodicity of sine, it's the same as $\beta' = \sqrt{2}$, which translates to $a = 2\pi \sqrt{2}$ from the OP. So, for this $a$ infimum is also the first term, which is also strictly positive.

Aleksei Kulikov
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