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.)
