Let $c>0$ be a very small constant and $N \in \mathbb N$ very large.
Assume we have a function $f(x)$ for $x \in S^1$ defined as
$$
f(x) = \sum\_{k=\lfloor N/(1+c) \rfloor}^{N} c\_k \sin(kx+b\_k)
$$
for some coefficients $b\_k, c\_k \in \mathbb R$, that is all the sines appearing have frequency similar to $N$.


I know that $|f(x)|\leq 1$ on $[-\pi/4, \pi/4]$. Can I get a good bound for arbitrary $x \in S^1$?


I'd love to get a bound independent of $N$ but I am not sure whether I can expect that.
I'd also love references where the authors treat these kinds of objects, but **I do not know what this kind of functions are called in the literature**.



Inequality for sums of sines with similar frequency
In other words, in particular, you want to create an entire function $f$ of exponential type $a=\pi/4$ bounded on the real axis by $C$ and satisfying $f(n)=(-1)^n$ near $N$ on an interval of length $cN$. Note that by symmetrizing ($F(z)=\frac 12(f(z)+\bar f(\bar z))$), you can make it real-valued on $\mathbb R$, and then it will have a zero between any two integers in that interval. Now Jensen in the disk of comparable radius kills all your hopes: the best bound you can get is still exponential in $N$.


