arXiv subjects

Nathaniel Kingsbury-Neuschotz

Publications and source records attributed to Nathaniel Kingsbury-Neuschotz.

4 recordsLinked to original sources

arXiv2026

Strong Approximation for the Relative Character Variety of the Four-Times Punctured Sphere

We study the orbits of solutions to the Markoff-type equation $$X^2 + Y^2 + Z^2 = XYZ + AX + BY + CZ + D$$ in $\mathbb{F}_p,$ for fixed integers $A, B, C, D,$ under the symmetry group $Γ$ generated by \[\begin{split}&V_1: (x, y, z)\mapsto (A + yz - x, y, z),\\ &V_2: (x, y, z)\mapsto (x, B + xz - y, z),\text{ and}\\ &V_3: (x, y, z)\mapsto (x, y, C + xy - z).\end{split}\] This equation defines the Relative Character Variety of the Four-Times Punctured Sphere, with $Γ$ arising from the Pure Mapping Class Group. Outside an explicit degeneracy locus, $Γ$ acts transitively on the bulk of solutions mod $p$ for density-one of primes, the remainder splitting into several small orbits reflecting finite orbits over $\mathbb{C}$. For the ``degenerate'' parameters, we show there are either two large orbits (most degenerate parameters) or four (the rest, excluding $(0, 0, 0, 4)$) for a density-one set of primes. These results are especially interesting for two subfamilies. The first, $$X^2 + Y^2 + Z^2 = XYZ + k,\,\,\,k\neq 4,$$ arises in the combinatorial group theory of $\text{SL}_2(\mathbb{F}_p)$; we very nearly prove the $Q$-classification conjecture of McCullough and Wanderley for density-one of primes. By work of Martin, this conjecture implies their Classification and $T$-Classification Conjectures. The second, $$x_1^2 + x_2^2 + x_3^2 + a_1x_2x_3 + a_2x_1x_3 + a_3x_1x_2 = (3+a_1+a_2+a_3)x_1x_2x_3,$$ arises from generalized cluster algebras. Our degeneracy notion specializes to that of de Courcy-Ireland, Litman, and Mizuno. For all nondegenerate and some degenerate surfaces in this subfamily, their results imply our orbit count (1, 2, or 4) holds for all sufficiently large primes.

math.NT↗

arXiv2026

On a Restriction Problem of Hickman and Wright for the Parabola over $\mathbb{Z}/N\mathbb{Z}$ for Squarefree $N$

Hickman and Wright proved an $L^2$ restriction estimate for the parabola $Σ$ over $\mathbb{Z}/N\mathbb{Z}$ of the form $$\left(\frac{1}{|Σ|}\sum\limits_{m\inΣ}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C_εN^ε\cdot N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{6}{5}\right)^\frac{5}{6}$$ for all functions $f:(\mathbb{Z}/N\mathbb{Z})^2\rightarrow \mathbb{C}$ and any $ε>0$, and showed that this bound is sharp when $N$ has a large square factor, especially for $N = p^2$ where $p$ is prime. In contrast, Mockenhaupt and Tao proved in the special case $N = p$ the stronger estimate $$\left(\frac{1}{|Σ|}\sum\limits_{m\inΣ}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{4}{3}\right)^\frac{3}{4}.$$ We extend the Mockenhaupt--Tao bound to the case of squarefree $N$, proving $$\left(\frac{1}{|Σ|}\sum\limits_{m\inΣ}|\widehat{f}(m)|^2 \right)^{\frac{1}{2}}\leq C_εN^ε\cdot N^{-1}\left(\sum\limits_{x\in (\mathbb{Z}/N\mathbb{Z})^2}|f(x)|^\frac{4}{3}\right)^\frac{3}{4},$$ and in fact a slightly sharper version with $C_εN^ε$ replaced with $2^\frac{ω(N)}{4}$, where $ω(N)$ is the number of prime factors of $N$. We also discuss applications of this result to uncertainty principles and signal recovery.

math.CA↗

arXiv2026

Square-Root Cancellation, Averages over Hyperplanes, and the Structure of Finite Rings

We formulate a notion of square-root cancellation for the operator which sums a mean-zero function over a rotating hyperplane in $R^d$, where $R$ is a possibly noncommutative finite ring. Using an argument due to Hart, Iosevich, Koh, and Rudnev, we show that this square-root cancellation occurs uniformly when $R$ is a finite field. We then show that this square-root cancellation cannot occur uniformly over families of finite rings which are not eventually finite fields. This extends an earlier result of the author to a non-translation-invariant operator.

math.NT↗

arXiv2025

The Square-Root Law Does Not Hold in the Presence of Zero Divisors

Let $R$ be a finite ring (with unit, not necessarily commutative) and define the paraboloid $P = \{(x_1, \dots, x_d)\in R^d|x_d = x_1^2 + \dots + x_{d-1}^2\}.$ Suppose that for a sequence of finite rings of size tending to infinity, the Fourier transform of $P$ satisfies a square-root law of the form $|\hat{P}(χ)|\leq C|R|^{-d}|P|^\frac{1}{2}$ for some fixed constant $C$ (for instance, if $R$ is a finite field, this bound will be satisfied with $C = 1$). Then all but finitely many of the rings are fields. Most of our argument works in greater generality: let $f$ be a polynomial with integer coefficients in $d-1$ variables, with a fixed order of variable multiplications (so that it defines a function $R^{d-1}\rightarrow R$ even when $R$ is noncommutative), and set $V_f = \{(x_1, \dots, x_d)\in R^d|x_d = f(x_1, \dots, x_{d-1})\}$. If (for a sequence of finite rings of size tending to infinity) we have a square root law for the Fourier transform of $V_f$, then all but finitely many of the rings are fields or matrix rings of small dimension. We also describe how our techniques let us see that certain varieties do not satisfy a square root law even over finite fields.

math.NT↗