Tight upper bounds on trigonometric polynomials

Question

According to D. Hajela's chapter in Open Problems in Communications and Computation the following question was open as of the late 1980s. I have been unable to find any references so any results or references will be appreciated:

Let $k$ be an arbitrarily large natural number. It is possible to show that there is a sequence of integers $(n_i)_{i\geq 1}$ such that for any $k\geq 1,$ the sequence has an initial segment $n_1,\ldots,n_k \in \mathbb{Z}$ satisfying $$ \sqrt{\pi k} \leq \max_{\theta \in (0,2\pi]} \left|\sum_{i=1}^k \sin n_i \theta\right|\leq k. $$ To see this, note that $$ \int_{0}^{2\pi} \left|\sum_{i=1}^k \sin n_i \theta\right|^2 = \pi k. $$ Along the same line, it is also known (Rudin-Shapiro polynomials) that there are $n_1,\ldots,n_k \in \mathbb{Z}\setminus \{0\}$ such that $$ \max_{\theta \in (0,2\pi]} \left|\sum_{i=1}^k \sin n_i \theta\right|\leq c\sqrt{k}. $$

(Open?) Problem from Hajela: For all arbitrarily large natural numbers $k$, are there $0<n_1\leq n_2\leq \cdots \leq n_k$ with $n_i$ integers such that $$ \max_{\theta \in (0,2\pi]} \left|\sum_{i=1}^k \sin n_i \theta\right|\leq c\sqrt{k}, $$ for some constant $c$?

Apparently it is known that $$ \max_{\theta \in (0,2\pi]} \left|\sum_{i=1}^k \sin n_i \theta\right|\leq ck^{2/3}, $$ but no reference to this is given.

(This open problem is stated to be due to H. Bohr from the 1950's, though no explicit reference is given.)

Answer 1

This is sometimes referred to as the "sine problem" in parallel with the much better known "cosine problem" of Chowla. I've seen the problem attributed to Bohr in connection with a problem about Dirichlet Series, but I've never seen a reference for that and I am not aware of what the application is. (Update: the paper of Konyagin cites the problem to Bohr's paper [3])

To state what is known, define

$S(K):= \min_{\{n_k\}_{k\in[K]}} \max_{\theta} |\sum_{1\leq k \leq K} \sin(n_k\theta) |$

where $\{n_k\}$ is an arbitrary set of $K$ natural numbers. Konyagin [1] has shown that:

$S(K) \gg K^{1/2} \left( \frac{\log K}{ \log \log K}\right)^{1/2}$

which gives a negative answer to the question as posed. In the other direction, Bourgain [2] constructed examples showing that $S(K) \ll K^{1/2 +1/6} = K^{2/3}.$

[1] S. V. Konyagin, Estimates of maxima of sine sums, East J. Approx. 3 (1997), no. 1, 63–7

[2] J. Bourgain, Sur les sommes de sinus, Harmonic analysis: study group on translationinvariant Banach spaces, Exp. No. 3, 9 pp., Publ. Math. Orsay 83, 1, Univ. Paris XI, Orsay, 1983.

[3] H. Bohr, A study of the uniform convergence of Dirichlet series and its connection with a problem concerning ordinary polynomails, Fys. Sall. Lund Forth 21 (1951) 103-118.