Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 11 / 3 / 2 / Solution

Write $\beta =|H|/M\gg {\delta }^8$ and $\eta =(\kappa /2{)}^{1/2}\gg {\delta }^4$. Choose unit complex numbers $b_h$ so thatis real and nonnegative. Cauchy-Schwarz, using $|F|\le 1$, bounds $T^2$ byAfter 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 yieldwhere $E_{\theta }(H)=\sum r(d,\xi {)}^2$ is the additive energy of a frequency graph. The last inequality uses ${\sum }_{\xi }|\widehat{{\partial }_{d}F}(\xi ){|}^2=\mathbb{E}|{\partial }_{d}F{|}^2\le 1$ for each $d$.

ThusThe 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 $C=48$ works for both conclusions, after adjusting absolute implicit constants and using $0<\delta \le 1$.

For the quadratic phase, the multiplicative derivative on the overlap is ${\partial }_h{f}_0(x)=e(2\alpha hx+\alpha h^2)$. Hence an explicit choice isTake $|h|\le \lfloor N/2\rfloor$. The correlation magnitude is $(N-|h|)/M\ge 1/32$, 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 functionIt has $\gg N^3$ exact additive quadruples, yet agrees with any affine function $ah+b(\mathrm{mod}1)$ 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 $\lfloor \sqrt{2}{h}_1\rfloor +\lfloor \sqrt{2}{h}_2\rfloor$ has only two possibilities, $\lfloor \sqrt{2}s\rfloor$ or $\lfloor \sqrt{2}s\rfloor -1$. 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 $(h,\lfloor \sqrt{2}h\rfloor )$ to be collinear: eliminating the affine slope gives $\sqrt{3}$ 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 $\lfloor \sqrt{2}h\rfloor$ to $\sqrt{2}h$ bounds their span by $1/|\sqrt{2}-p/q|$. Lines with $q$ large have arbitrarily small possible density; for bounded $q$, irrationality of $\sqrt{2}$ bounds the number uniformly. Thus the maximum agreement is $o(N)$.