My question concerns $L^p ([0,1]^d)$ estimates for trigonometric polynomials, where both the coefficients and frequencies are coming from general (i.e. not necessarily geometrically special/structured) sets. I give the set-up, then ask the question (bolded). Thank you for reading!
Suppose that $f: \mathbb{R}^d \rightarrow \mathbb{C}$ is some trigonometric polynomial $$ (*) \quad f(x) : = \sum\limits_{a \in A} c_a e^{2 \pi i a \cdot x} $$ with associated frequency set $A \subset \mathbb{Z}^d$ and coefficients $c_a \in \mathbb{C}$. We assume that $\# A = M < \infty$, so that $A$ has finite cardinality. Also, for simplicity, let's assume that $A$ contains only non-negative vectors, so that $A \subset \{0,...,N-1\}^d$ for some $N \in \mathbb{N}$.
I am interested in the sort of norm estimates,
\begin{equation}\label{exponential} (**) \quad \vert \vert f(x) \vert \vert_{L^p ([0,1]^d)} \leq D_{p,M,N} \bigg(\sum\limits_{a \in A} \vert c_a \vert^2 \bigg)^{\frac{1}{2}}, \quad p > 2, \end{equation} Particularly, I am curious in estimation of the constant $D = D_{p,M,N}$ as a function of the Lebesgue exponent $p > 2$, the cardinality $M$ of the frequency set $A$ and the parameter $N$.
Using Minkowski's integral inequality, together with Cauchy-Schwartz, it is rather easy to show that $D_{p,M,N}$ and be taken $\leq M^{1/2}$, for any $p > 2$. But this is an elementary bound, and should be lossy, especially since Minkowski's inequality is inherently lossy for a series of exponential functions.
On the other end, J. Bourgain showed (in Bounded Orthogonal Systems...) that, given $N$ large enough, there always exists some $A \subset \{0,...,N-1\}^d$ satisfying $\# A = M \sim N^{\frac{2d}{p}}$ such that $(**)$ holds for all sequences $c_{a}$ and with $D_{p,M,N} = D_p$ only depending upon the Lebesgue exponent $p > 2$. Such sets are called $\Lambda(p)$ sets. But these are highly structured sets (from the view of additive combinatorics).
My Question: Given $1 \ll M \ll N$ (so, we are thinking of $M$ as large, but with $N$ as much larger than $M$) does there exist a general upper bound of type $(**)$ which holds for all sets $A \subset \{0,...,N-1\}^d$ satisfying $|A| \sim M$, and for which the constant $D_{p,M,N}$ is stronger than the Cauchy-Schwartz estimate? So, something like \begin{equation} (***) \quad \vert \vert f(x) \vert \vert_{L^p ([0,1]^d)} \leq M^{1/2 - \epsilon_{p}} \bigg(\sum\limits_{a \in A} \vert c_a \vert^2 \bigg)^{\frac{1}{2}}, \quad p > 2, \end{equation} for some quantifiable $\epsilon_p > 0$, which of course depends upon the choice of $p > 2$? If so, I would appreciate a reference (or maybe it's just a simple trick or calculation I am missing). Otherwise, if this is false, I would very much appreciate a counterexample.
Thank you for your time reading!
