Popular square differences: a radix obstruction and a sharp characteristic-two no-wrap model

Definitions and mathematical statement

For $A\subseteq[N]=\{1,\ldots,N\}$, put $\alpha=|A|/N$ and $P_A(y)=|\{x\in\mathbb Z:x,x+y^2\in A\}|$. For $0<\varepsilon<1$, define $N_0(\varepsilon)$ as the least integer $M$ such that every $N\geq M$ and every $A\subseteq[N]$ admit a nonzero integer $y$ satisfying \[ P_A(y)\geq(\alpha^2-\varepsilon)N. \] The integer question asks for strong lower bounds on this threshold: in particular, can one prove growth faster than every fixed power of $\varepsilon^{-1}$, or already the bound $N_0(\varepsilon)>\varepsilon^{-10}$ for all sufficiently small $\varepsilon$? A lower bound requires examples in which every nonzero square difference has too few representations, rather than just one unpopular difference.

There is also a function-field version. For a fixed prime $p$, let $V_n=\{f\in\mathbb F_p[t]:\deg f<n\}$, of size $p^n$. Define $n_{0,p}(\varepsilon)$ by the same universal property for $A\subseteq V_n$, with density $\alpha=|A|/p^n$, requiring a nonzero polynomial $y$ with $\deg(y^2)<n$ and \[ |\{x\in V_n:x,x+y^2\in A\}|\geq(\alpha^2-\varepsilon)p^n. \] If no eventual threshold exists, assign it the value $\infty$. Determine upper and lower bounds, allowing dependence on the fixed characteristic. These are ordinary polynomial differences, without reduction modulo a polynomial; characteristic two is included as a potentially different case.

Short English statement

How large must an interval, or a polynomial vector space, be before every subset has an almost density-squared number of pairs at some nonzero square difference?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000884 (AIM-COMBINATORICS-0009), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, High-dimensional phenomena in discrete analysis, item 2.6. Original workshop question underlying AIM-COMBINATORICS-0009. http://aimpl.org/highdimdiscrete/2/.