arXiv subjects

Connor Yau

Publications and source records attributed to Connor Yau.

2 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↗