An observation of Nazarov on $\Lambda(p)$-systems I have been reading this note of F. Nazarov "Summability of large powers of logarithm of classic lacunary series and its simplest consequences (1995)", available here https://users.math.msu.edu/users/fedja/prepr.html.
At the end of page 3, he makes a claim about $\Lambda(p)$ systems, which I struggle to prove.  I will now introduce some background and explain the said claim.
Background
Let $\mathbb{T}$ be the unit circle. Given a function function $f\in L^2(\mathbb{T})$, the spectrum of $f$ is
$Spec(f)= \{n\in \mathbb{Z}: \hat{f}(n)= \int_{\mathbb{T}} f(n) \exp(2\pi i x)dx\neq 0\}.$
Let $p>2$ a positive integer. A set $V\subset \mathbb{Z}$ is a $\Lambda(p)$ systems if for every function $f\in L^2(\mathbb{T})$ with $Spec(f)\subset V$, we have
$||f||_{L^p}\leq A(V,p)||f||_{L^2},$
for some constant $ A(V,p)>0$.
We need a last piece of notation, given $V\subset \mathbb{Z}$, we write
$D(V):=\{n_i-n_j: i\neq j, n_i,n_j\in V\}$
for the set of differences minus the diagonal.
The claim is the following. Let $p>2$ be an integer. Suppose that there are no non-trivial linear combinations with integer cofficents of sum of absolute values less than 2p among the elements of $V=\{n_i\}\subset \mathbb{Z}$, that is
$\sum_i a_i n_i=0 \hspace{5mm} \sum|a_i|\leq 2p \hspace{5mm} a_i\in \mathbb{Z}$  then $a_i=0$ for all i.
Then $D(V)$ is a $\Lambda(p)$ systems. Moreover, I think that the constant A in the definition should be independet of $D(V)$.
Observations
I know that the assumption of the claim implies that $V$ is a $\Lambda(2p)$ system with constant A, independent of $V$ (just by expanding the $L^{2p}$-norm and using this weak $\mathbb{Z}$ linear independence). I guess that the proof of the claim is similar. However, expanding the $L^p$-norm, we find solutions of the following form (for p=4 say) given $a,b,c,d\in V$ we can take $m_1=a-b$, $m_2=c-b$, $m_3= c-d$ and $m_4= a-d$, then $m_1-m_2+m_3-m_4=0$, and I am not sure how to bound their contribution.
 A: Actually, you take it from a wrong end. It is much easier to understand another claim made there, which is that the $\Lambda_p$-constant for such spectrum is the least possible for a difference set (note, by the way, that the claim in the paper was made for even integers $p$ only). Indeed, if all coefficients are positive, then the terms corresponding to forced algebraic relations are appearing in the expansion of the $L^p$-norm for every difference set spectrum and there may be more terms there. Now you know that difference sets that are $\Lambda_p$ systems do exist  (the difference set of any set with Hadamard gaps is such), and that is it.
Now, of course, you may want a direct proof. Then just think of how you use the Littlewood-Paley decomposition for the lacunary case (and how you prove that decomposition for even integer powers without resorting to singular integral operator techniques) and everything will fall into place. You'll see that you will just need one intermediate step, which is the decomposition $f=\sum_{k=1}^\infty f_k$ where $f_k$ takes in all frequencies $n_k-n_j$ with $j=1,2,\dots,k-1$ and then you can just mimic the proof of the Littlewood-Paley inequality
$$
\int |f|^p\le C_p\int\left[\sum_k |f_k|^2\right]^{p/2}
$$
after which the story ends exactly the same way as for the spectrum with Hadamard gaps.
