Nathaniel Kingsbury-Neuschotz

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20242026

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math.NT2026

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

Nathaniel Kingsbury-Neuschotz

We study the orbits of solutions to the Markoff-type equation $X^{2} + Y^{2} + Z^{2} = X Y Z + A X + B Y + C Z + D$ in $F_{p},$ for fixed integers $A, B, C, D,$ under the symmetry gr…

math.NT2025

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

Nathaniel Kingsbury-Neuschotz

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 finit…

math.NT2024

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

Nathaniel Kingsbury-Neuschotz

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 s…