arXiv subjects

Joshua Khan

Publications and source records attributed to Joshua Khan.

4 recordsLinked to original sources

arXiv2026

Random Fibonacci Tilings

The Fibonacci spiral is formed by a square-joining rule: begin with a unit square and attach squares whose side lengths follow the Fibonacci sequence; in the standard construction, choices of direction produce a spiral and a tiling of the plane. We study a randomization of this construction, in which the attachment axis alternates deterministically, while fair coin flips determine direction. For a lattice point we ask when the growing rectangle first covers that point and how the expected covering time depends on its location. Tracking the rectangle's boundary reduces this question to random subset sums of even- and odd-indexed Fibonacci numbers. We extend the construction to d dimensions, where coordinate subsequences are super-increasing, allowing us to derive exact covering probabilities and formulas for the expectation and variance. For $d\geq2$, the expected covering time grows logarithmically with distance from the initial block, while the variance remains uniformly bounded. We obtain exponential tail bounds for the delay beyond the earliest feasible covering stage. Finally, we extend the Fibonacci spiral to the randomized tiling through a square-by-square construction using circular arcs and quintic connectors. This construction uses one-step lookahead, admits a continuously differentiable parameterization, and recovers the Fibonacci spiral for the standard tiling.

math.PR↗

arXiv2026

Breaking Universality in the Lower Order Terms in the 1-level and 2-level Density of Holomorphic Cusp Newforms

The Katz--Sarnak density conjecture predicts that the normalized low-lying zeros of a family of $L$-functions ordered by analytic conductor $R$ converge, as $R \to \infty$, to the eigenvalue statistics near $1$ of a classical compact group, and that the limit depends on the family only through its symmetry type. We study the first correction to this limit for families of holomorphic cusp newforms of fixed weight $k$ and varying level $N$. Sharpening work of Miller on lower order terms in the one-level density, we evaluate the harmonically weighted one- and two-level densities with errors $O\left(\log^{-5}(R)\right)$ and $O\left(\log^{-2}(R)\right)$, and show that each of them already differs at order $\log^{-1}(R)$ from an explicit universal profile built out of $k$, the test functions, and $R$ alone. We compute that difference exactly. It is the product of a functional of the test functions with a single arithmetic invariant $κ_1(F)$ of the family, an explicit sum over the bounded prime factors of the level. Two families of newforms of the same weight and the same symmetry type, with conductors of the same size, therefore have densities whose first lower order terms differ by an explicit nonzero amount whenever their levels have different bounded prime factors.

math.NT↗

arXiv2026

Product and Quotient Set Dominance in Finite Groups

A classic problem in additive number theory asks whether a given subset of integers has more sums or differences. This inquiry naturally extends to finite groups, where one compares the sizes of the product set $AA$ and the quotient set $AA^{-1}$ for a subset $A \subseteq G$. A finite group is considered product-dominant if it contains a higher proportion of subsets where $|AA| > |AA^{-1}|$, and quotient-dominant if the reverse is true. We establish statistical criteria to determine the dominance behavior of large finite groups by analyzing the moments of the random variable $d(A) := |AA| - |AA^{-1}|$. By translating product and quotient relationships into graph-theoretic constructions, we compute these moments through counting independent sets. Our main result demonstrates that for a growing sequence of finite groups $G_k$ with center size $Z_k$ and involution count $T_k$, the condition $Z_k = o(T_k)$ implies the groups are eventually product-dominant. Conversely, if $T_k \cdot \sqrt{3}^{R_k/Z_k} = o(Z_k)$ (where $R_k$ denotes the number of elements whose square is central), the groups are eventually quotient-dominant. Using these asymptotic conditions, we resolve the Miller--Vissuet conjecture by proving that all dihedral groups $D_n$ for $n \geq 3$ are product-dominant. Furthermore, we classify the dominance behavior of several major families of groups, proving that symmetric groups and non-abelian finite simple groups are product-dominant, while groups of odd order and certain abelian groups are quotient-dominant. Finally, for groups that fall outside our asymptotic conditions, we derive an explicit formula for the first moment utilizing the irreducible characters and conjugacy classes of the group.

math.NT↗

arXiv2026

Bounding The Number of Zeros Near the Central Point in Families of Cuspidal Newforms

We study low-lying zeros in families of even holomorphic cuspidal newforms of fixed weight and prime level, with particular emphasis on the number of forms having a zero in a prescribed normalized window about the central point and on the distribution of the number of such zeros among the forms. We quantify the number of forms having at least one zero in the window and study the distribution of the number of zeros in that window among the forms. Assuming the Generalized Riemann Hypothesis, we obtain new lower bounds for the number of forms having a low-lying zero. We first prove that, along an infinite sequence of prime levels $N$, the number of such forms is $\gg N^{7/8}\log N$. We then use higher centered moments and an appropriate test function to show that a positive proportion of the family has a zero in a prescribed window. Finally, we obtain polynomial upper-tail bounds for the number of zeros occurring there and show that a positive proportion of forms have a bounded, nonzero number of low-lying zeros.

math.NT↗