4 papers
Periodic Tilings of Cardinality Twice a Prime or Nine in Arbitrary Dimension
Hu Tan, Ying Zhang
We prove that every finite translational tile of of cardinality , where is an odd prime, or of cardinality nine admits a fully periodic tiling complement. The…
Sharp Diameter Bounds for Nonnegative Cyclotomic Multiples
Hu Tan, Ying Zhang
Let \(N\ge2\) and let \(p\) be its least prime divisor. We prove that every nonzero polynomial with nonnegative real coefficients divisible by \(Φ_N\) has support diameter at least…
Unsupported Cyclotomic Divisors in Three-Prime Integer Tilings
Hu Tan, Ying Zhang
Cyclotomic divisibility imposes strong prime-power structure on integer tiles. We study unsupported cyclotomic divisors: mixed-order divisors for which none of the prime-power comp…
A Proof of Fraenkel's Conjecture
Hu Tan, Ying Zhang
Fraenkel's conjecture asserts that a partition of the integers into at least three Beatty sequences with distinct moduli has the binary densities , normalized…