$L_p$ norms of $0-1$ exponential sums

Question

Consider $f_n(t)=\sum_{i=1}^{n}e^{ik_{i}t}$ with all $k_i$ some distinct integers for $t\in [-\pi,\pi)$. For $p>2$ I am interested in the maximum possible value of $$||f_n||_p,$$ where $f_n$ runs through all possible such sums with $n$ terms. Of particular interest is the case when $p$ is an even integer and $n\rightarrow \infty$ in case obtaining the sharp upper bound is too ambitious. Could it be that the Dirichlet kernel is the best such $f_n$?

Answer 1

For even integer exponents, say $p=2k$ and $p \geq2$, the quantity is just the $k$-order additive energy of the set $S \subset \mathbb{Z}$ of non-zero Fourier coefficients. It is easy to see that this is maximized by any arithmetic progression of the desired length (which coincides with the $n$-order Dirichlet kernel).

In the case of $p=4$ this is just the statement that an arithmetic progression maximizes the quantity $$E(A) := \sum_{\substack{a,b,c,d \in A \\a+b=c+d}}1$$ of all integer subsets of cardinality $|A|$, which is easy to see.

For $p<2$ arithmetic progressions should minimize the quantity. This is a hard and difficult unsolved problem.