Let $(n_k)_{1 \leq k \leq N}$ be a sequence of distinct positive integers. For $v \in \mathbb{Z}$ set $$ A_N(v) = \# \Big\{ (k,\ell) \in \{1, \dots, N\}^2, ~k \neq \ell:\quad n_k - n_\ell = v \Big\}. $$

Assume that $$ \sum_{v \in \mathbb{Z}} A_N(v)^2 \geq \varepsilon N^{3} $$ for some $\varepsilon > 0$. (Assume that $\varepsilon$ is fixed and $N$ is "very large".)

Question: Does the sequence $(n_k)$ necessarily have a strong arithmetic substructure? In particular, does there necessarily exist a number $w$ such that all the numbers $$ A_N(w), \quad A_N(2 w), \quad A_N(M w) $$ for some "large" value of $M$ all are "large"? (If so, please provide something quantitative, such as: $A_N(h w) \geq const(\varepsilon)N$ for all $h \leq H(N)$ for a certain function $H$.)

(Remarks: the natural example of a sequence for which $\sum A_N(v)^2$ is large is the sequence $(1, 2, \dots, N)$. For this sequence we clearly have the desired phenomenon. Note that the given lower bound for $\sum A_N(v)^2$ is quite strong, since trivially $\sum A_N(v)^2 \leq N^3$ by the fact that all $n_k$'s are different.)