Ying Zhang

Uniqueness of Markov numbers with a large prime-power divisor

The Frobenius uniqueness conjecture asserts that the largest entry of a positive integral solution of a^2 + b^2 + m^2 = 3abm determines the other two entries up to order. We prove that a Markov number m > 2 is unique if it has an odd prime-power divisor q satisfying ε q > 3^{-6/4...

Uniform cubic bounds for irrational values of the dilogarithm

We study explicit denominator conditions ensuring the irrationality of the dilogarithm at positive and negative rational points. We prove that Li_2(p/q) is irrational whenever q > (4e^{4}/27)p^3, and obtain an explicit, slightly smaller threshold for Li_2(−p/q). The positive argu...

A dual-cubic proof of the four-point hyperbolic Atiyah conjecture

The hyperbolic Atiyah conjecture assigns a binary form of degree n-1 to each point of an n-point configuration in hyperbolic three-space and asserts that the resulting forms are linearly independent. We give a complete geometric proof for four points. Starting from Malkoun's dual...

Counterexamples to the Knaster problem in all remaining dimensions

We construct counterexamples to Knaster's 1947 problem in all remaining cases. For every ambient dimension n ≥ 4, we obtain a real polynomial on R^{n} and n linearly independent points on S^{n-1} whose values cannot be equalized by any orthogonal transformation. These examples se...

The Atiyah-Sutcliffe determinant at five points: positivity and failure of the deletion inequality

To investigate Atiyah's linear independence conjecture, Atiyah and Sutcliffe associated a normalized determinant to each configuration of distinct points in Euclidean 3-space and proposed a hierarchy of nonvanishing and determinant inequalities. We show that this hierarchy separa...

Density of Triple Coincidences in the Simple Length Spectrum of One-holed Tori

We prove that, in the Teichmüller space of one-holed tori with any fixed boundary length, including the cuspidal case, marked hyperbolic one-holed tori with three distinct simple closed geodesics of equal length form a dense, meager subset. This answers a suspicion of McShane and...

Density of Triple Coincidences in the Simple Length Spectrum of One-holed Tori

We prove that, in the Teichmüller space of one-holed tori with any fixed boundary length, including the cuspidal case, marked hyperbolic one-holed tori with three distinct simple closed geodesics of equal length form a dense, meager subset. This answers a suspicion of McShane and...

Cohn Trace Polynomials: Root Localization, Strict Log-Concavity, and Rearrangement Inequalities

We study the one-variable trace polynomials obtained from words in the rank-two free group by the Cohn normalization. We prove that every nontrivial word has all its zeros in a fixed sector about the negative real axis; after the marking involution, this is equivalently a transla...

Fuglede's conjecture for cyclic groups of square-free order

We prove Fuglede's conjecture for finite cyclic groups of square-free order. Specifically, if N is square-free and A is a nonempty subset of Z_{N} with |A|=m, then A is spectral if and only if it tiles Z_{N}; moreover, these conditions hold precisely when m divides N and A is a c...

Fuglede's conjecture for cyclic groups of square-free order

We prove Fuglede's conjecture for finite cyclic groups of square-free order. Specifically, if N is square-free and A is a nonempty subset of Z_{N} with |A|=m, then A is spectral if and only if it tiles Z_{N}; moreover, these conditions hold precisely when m divides N and A is a c...

Periodic joint co-tiles of independent tiles

For every d ≥ 2, we prove that d-1 independent finite tiles in Z^{d} admit a fully periodic joint co-tile whenever they admit a joint co-tile. Here independence means that the tiles contain the origin and every choice of one nonzero vector from each tile is linearly independent.

Periodic joint co-tiles of independent tiles

For every d ≥ 2, we prove that d-1 independent finite tiles in Z^{d} admit a fully periodic joint co-tile whenever they admit a joint co-tile. Here independence means that the tiles contain the origin and every choice of one nonzero vector from each tile is linearly independent.