Exact horizontal Plancherel identities for the diagonal Diffie--Hellman sum

Definitions and mathematical statement

Let $p$ be prime, $\theta$ a generator of ${\mathbb{F}}_p^*$, $a,b\in{\mathbb{F}}_p$, and $e_p(z)=\exp(2\pi i z/p)$. J. Bourgain's source asks for a good upper bound on a sum extracted as $\sum_{x\in{\mathbb{F}}_p}e_p(a\theta^x+b\theta^{x^2})$. Exponents of $\theta$ naturally live modulo $p-1$, so an index $x\in{\mathbb{F}}_p$ is not canonical without integer representatives. The coefficient restrictions and desired saving are also unspecified; $(a,b)=(0,0)$ precludes a nontrivial bound valid for all coefficients.

The attached aid distinguishes the periodic and representative conventions: \[ T(a,b)=\sum_{x=0}^{p-2}e_p(a\theta^x+b\theta^{x^2}),\qquad S(a,b)=\sum_{x=0}^{p-1}e_p(a\theta^x+b\theta^{x^2}). \] The first is a sum over ${\mathbb{Z}}/(p-1){\mathbb{Z}}$, and the second fixes the usual representatives of ${\mathbb{F}}_p$. Their exact endpoint relation is \[ S(a,b)=T(a,b)+e_p(a+b). \] For every fixed $b$, the aid gives the exact horizontal second moments \[ \sum_{a\in{\mathbb{F}}_p}|T(a,b)|^2=p(p-1),\qquad \sum_{a\in{\mathbb{F}}_p}|S(a,b)|^2=p(p+2). \] Consequently, for every $H>0$, \[ \#\{a:|T(a,b)|>H\sqrt p\}\le\frac{p-1}{H^2},\qquad \#\{a:|S(a,b)|>H\sqrt p\}\le\frac{p+2}{H^2}. \] These bounds are uniform in the fixed second coefficient but average over the first; they do not give a square-root bound for every coefficient pair or identify the exceptional coefficients.

The coefficient line $b=0$ is evaluated exactly: \[ T(0,0)=p-1,\quad S(0,0)=p,\qquad T(a,0)=-1,\quad S(a,0)=e_p(a)-1\quad(a\ne0). \] The general worst-case pointwise cancellation requested by the source remains outside these moment and endpoint identities.

Short English statement

Bound the mixed exponent sum after fixing its index convention. The supplied aid gives exact second moments and exceptional-set estimates in one coefficient, plus the zero-second-coefficient line, rather than a worst-case pointwise theorem.

Sources

• Local UnsolvedMath registry, permanent ID $20001130$, source code catalogue entry; canonical aim-combinatorics-notes.json, notes[254].
• American Institute of Mathematics, Recent trends in additive combinatorics, problem 3.18 attributed to J. Bourgain: https://aimath.org/WWN/additivecomb/additivecomb.pdf. The index and superscripts are extraction damaged; no original remarks or literature references are supplied. No PDF inspection is claimed here.
• Attached research attempt 1: partial result, recorded run status unknown, rigor proved, novelty confidence low; registry status partially solved, classification PARTIAL-PROGRESS. Its endpoint-aware identities are machine generated and not peer reviewed; independently verify before citation. No authoritative pointwise-bound reference is supplied.