= Solution
Write $\beta=|H|/M\gg\delta^8$ and $\eta=(\kappa/2)^{1/2}\gg\delta^4$. Choose unit complex numbers $b_h$ so that
$$
T=\mathbb E_x\overline{F(x)}\left(\frac1M\sum_{h\in H}b_hF(x+h)e(-\theta(h)x)\right)\geq\beta\eta
$$
is real and nonnegative. <Cauchy-Schwarz>, using $|F|\leq1$, bounds $T^2$ by
$$
\frac1{M^2}\sum_{h,k\in H}b_h\overline{b_k}\,\mathbb E_xF(x+h)\overline{F(x+k)}e(-[\theta(h)-\theta(k)]x).
$$
After setting $y=x+k$, each summand is a unit phase times the <Fourier coefficient on a finite abelian group> of $\partial_{h-k}F$ at frequency $\theta(h)-\theta(k)$.
Let $r(d,\xi)$ count pairs $(h,k)\in H^2$ with $h-k=d$ and $\theta(h)-\theta(k)=\xi$. Grouping terms, another <Cauchy-Schwarz> and the <Parseval identity> yield
$$
T^2\leq\frac1{M^2}\left(\sum_{d,\xi}r(d,\xi)^2\right)^{1/2}
\left(\sum_{d,\xi}|\widehat{\partial_dF}(\xi)|^2\right)^{1/2}
\leq\left(\frac{E_\theta(H)}{M^3}\right)^{1/2},
$$
where $E_\theta(H)=\sum r(d,\xi)^2$ is the <additive energy of a frequency graph>. The last inequality uses $\sum_\xi|\widehat{\partial_dF}(\xi)|^2=\mathbb E|\partial_dF|^2\leq1$ for each $d$.
Thus
$$
\boxed{E_\theta(H)\geq(\beta\eta)^4M^3\gg\delta^{48}N^3.}
$$
The bound proves that <derivative correlations force additive frequency energy>. This energy counts exactly the ordered quadruples satisfying the two requested additive relations, after relabelling the difference equality as a sum equality. Shift equalities initially hold modulo $M$, but the shifts lie in $[-N,N]$ and $M>8N$, so they are also integer equalities. Frequency equalities hold in $\mathbb R/\mathbb Z$, as required. Consequently a common exponent \b[$C=48$] works for both conclusions, after adjusting absolute implicit constants and using $0<\delta\leq1$.
For the <quadratic phase>, the <multiplicative derivative> on the overlap is $\partial_hf_0(x)=e(2\alpha hx+\alpha h^2)$. Hence an explicit choice is
$$
\boxed{\theta(h)=2\alpha h\pmod1.}
$$
Take $|h|\leq\lfloor N/2\rfloor$. The correlation magnitude is $(N-|h|)/M\geq1/32$, and every additive quadruple of shifts satisfies the frequency relation. The interval of shifts has $\gg N^3$ such quadruples by <Cauchy-Schwarz>. These particular real frequencies need not belong to the grid used to prove existence for general $f$; evaluating them at the interval's integer coordinates gives the stated exact formula.
A genuinely nonlinear example is the <bracket-linear frequency function>
$$
\boxed{\theta(h)=\sqrt3\,\{\sqrt2\,h\}\pmod1,\qquad H=[-N,N]\cap\mathbb Z.}
$$
It has $\gg N^3$ exact additive quadruples, yet agrees with any affine function $ah+b\pmod1$ at only $o(N)$ shifts, uniformly in the affine function. The question permits giving this example without proof, but the mechanism is useful. For a pair with sum $s$, the value $\lfloor\sqrt2h_1\rfloor+\lfloor\sqrt2h_2\rfloor$ has only two possibilities, $\lfloor\sqrt2s\rfloor$ or $\lfloor\sqrt2s\rfloor-1$. There are $O(N)$ pair classes, so <Cauchy-Schwarz> produces $\gg N^3$ collisions with both additive relations.
To see why no long affine agreement occurs, three agreement points force the corresponding lattice points $(h,\lfloor\sqrt2h\rfloor)$ to be collinear: eliminating the affine slope gives $\sqrt3$ times an integer determinant equal to an integer, and irrationality makes that determinant zero. A rational line of slope $p/q$ contains agreement shifts in one residue class modulo $q$, and closeness of $\lfloor\sqrt2h\rfloor$ to $\sqrt2h$ bounds their span by $1/|\sqrt2-p/q|$. Lines with $q$ large have arbitrarily small possible density; for bounded $q$, irrationality of $\sqrt2$ bounds the number uniformly. Thus the maximum agreement is $o(N)$.
Back to article page
