Heilbronn sums as exact Fourier quadrature

Definitions and mathematical statement

Original question (Katz). The extracted Heilbronn sums appear, as the parameter $t$ varies, to approximate a pushforward of Haar measure better than expected. Why? The displayed identity is damaged and, according to the aid, literally loses the summation-variable dependence in its final exponential. The source does not specify $t$'s range, normalization, compact group, pushforward map or meaning of "better than expected".

Explicit aid reconstruction, not certified as Katz's intended Haar target. Write $e_M(u)=\exp(2\pi i u/M)$. For an odd prime $p$, the algebraically corrected family is \[S_p(t)=\sum_{x\bmod p}e_{p^2}\left(x^p(1+pt)\right)=\sum_{x\bmod p}e_{p^2}(x^p)e_p(tx),\qquad t\in{\mathbb{F}}_p.\] The second factor depends on $x$. The expression $x^p\bmod p^2$ is independent of its lift from ${\mathbb{F}}_p$, and the family is periodic modulo $p$ in $t$. This reconstruction makes the lost dependence explicit rather than accepting the false printed product.

Put $h=(p-1)/2$, $I_p=\{-h,\ldots,-1,1,\ldots,h\}$ and \[F_p(z)=\sum_{x\in I_p}e_{p^2}(x^p)z^x,\qquad Y_p(z)=\frac{F_p(z)}{\sqrt{p-1}}.\] For $|z|=1$ these are real, and $S_p(t)=1+F_p(e_p(t))$. The chosen centered normalization has Haar second moment one. Let $\mu_p$ be the empirical measure of $Y_p(e_p(t))$ for uniform $t\in{\mathbb{F}}_p$, and $\nu_p=(Y_p)_*m_{S^1}$, where $m_{S^1}$ is normalized circle Haar measure. This is a comparison to a polynomial-dependent circle pushforward, not an identified random-matrix target or a measure fixed independently of $p$.

For every integer $m\ge1$, the aid's wraparound reduction is \[\int y^m\,{\,\mathrm{d}}\mu_p-\int y^m\,{\,\mathrm{d}}\nu_p=(p-1)^{-m/2}\sum_{\substack{x_1,\ldots,x_m\in I_p\\x_1+\cdots+x_m=kp,\ k\ne0}}e_{p^2}(x_1^p+\cdots+x_m^p).\] For nonzero $x\in I_p$ define the Fermat quotient $q_p(x)=(x^{p-1}-1)/p\pmod p$. On the slice with sum $kp$, its phase equals \[e_p\left(k+\sum_jx_jq_p(x_j)\right).\] The aid reports exact agreement of the first two centered moments. For the cubic moment the entire difference becomes \[\frac{2}{(p-1)^{3/2}}\operatorname{Re}\sum_{\substack{1\le x,y,z\le h\\x+y+z=p}}e_p\left(1+xq_p(x)+yq_p(y)+zq_p(z)\right).\] The triangle is empty when the displayed constraints have no solutions. The aid leaves the cancellation required in this positive triangular Fermat-quotient sum open. It supplies no quantitative target strong enough to formalize "better than expected" as a proved rate. The odd-prime restriction and centered normalization are explicit choices of this reconstruction; the original gives neither and does not discuss $p=2$.

Short English statement

The printed identity and Haar comparison are underspecified. The aid corrects the missing variable factor and models the sums as circle sampling of a real Laurent polynomial. Moment error then comes exactly from Fourier frequencies wrapping by a nonzero multiple of $p$; the first two centered moments agree, while the cubic error remains a cancellation problem.

Sources

• Local UnsolvedMath registry, permanent ID 20000400, source code catalogue entry; aim-algebraic-number-theory-notes.json, notes[51], tag question.
• AIM workshop Arithmetic statistics over finite fields and function fields, Question 22 (Katz): https://aimath.org/pastworkshops/arithstatffieldproblems.pdf. The record is reconstructed_unverified, with no clean statement. No PDF was consulted.
• The supplied verification explicitly says the missing compact group and map were not verified as Katz's intended wording and that a different random-matrix target cannot be recovered from the PDF alone.
• Attached machine-generated research attempt 1: result partial_result, problem status at run open, rigor label proved, novelty confidence low. It is not peer reviewed and requires independent verification before citation. No authoritative bibliography is supplied.