60 citations · 281 across the 30 of their papers we have counts for
31 papers
Higher uniformity of arithmetic functions in short intervals I. All intervals
Kaisa Matomäki, Xuancheng Shao, Terence Tao +1
We study higher uniformity properties of the Möbius function , the von Mangoldt function , and the divisor functions on short intervals with $X^{θ+\varepsilon…
Upperbound for Dimension of Hilbert Cubes contained in the Quadratic Residues of
Ali Alsetri, Xuancheng Shao
We consider the problem of bounding the dimension of Hilbert cubes in a finite field that does not contain any primitive roots. We show that the dimension of such Hi…
Singmaster's conjecture in the interior of Pascal's triangle
Kaisa Matomäki, Maksym Radziwiłł, Xuancheng Shao +2
Singmaster's conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal's triangle; that is, for any natural number …
A transference principle for systems of linear equations, and applications to almost twin primes
Pierre-Yves Bienvenu, Xuancheng Shao, Joni Teräväinen
The transference principle of Green and Tao enabled various authors to transfer Szemerédi's theorem on long arithmetic progressions in dense sets to various sparse sets of integers…
The Bombieri-Vinogradov theorem for nilsequences
Xuancheng Shao, Joni Teräväinen
We establish results of Bombieri-Vinogradov type for the von Mangoldt function twisted by a nilsequence. In particular, we obtain Bombieri-Vinogradov type results for the vo…
Discorrelation between primes in short intervals and polynomial phases
Kaisa Matomäki, Xuancheng Shao
Let and . We obtain estimates for the following exponential sum over primes in short intervals: \[ \sum_{N < n \leq N+H} Λ(n) e(g(n)), \] where is a…