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## Vinogradov's theorem with almost equal summands

- Kaisa Matomäki, J. Maynard, X. Shao
- Mathematics
- 6 October 2016

Let θ>11/20 . We prove that every sufficiently large odd integer n can be written as a sum of three primes n=p1+p2+p3 with |pi−n/3|⩽nθ for i∈{1,2,3} .

## On character sums and exponential sums over generalized arithmetic progressions

- X. Shao
- Mathematics
- 4 June 2012

Let χ (mod q) be a primitive Dirichlet character. In this paper, we prove a uniform upper bound of the character sum ∑a∈Aχ(a) over all proper generalized arithmetic progressions A⊂ℤ/q ℤ of rank r:… Expand

## Bombieri-Vinogradov for multiplicative functions, and beyond the x1/2-barrier

- A. Granville, X. Shao
- MathematicsAdvances in Mathematics
- 20 March 2017

## Vinogradov’s three primes theorem with almost twin primes

- Kaisa Matomäki, X. Shao
- MathematicsCompositio Mathematica
- 10 December 2015

In this paper we prove two results concerning Vinogradov’s three primes theorem with primes that can be called almost twin primes. First, for any $m$ , every sufficiently large odd integer $N$ can be… Expand

## Polynomial values modulo primes on average and sharpness of the larger sieve

- X. Shao
- Mathematics
- 25 September 2014

This paper is motivated by the following question in sieve theory. Given a subset $X\subset [N]$ and $\alpha\in (0,1/2)$. Suppose that $|X\pmod p|\leq (\alpha+o(1))p$ for every prime $p$. How large… Expand

## Type-IV DCT, DST, and MDCT algorithms with reduced numbers of arithmetic operations

- X. Shao, S. Johnson
- Computer ScienceSignal Process.
- 31 August 2007

## On an almost all version of the Balog-Szemeredi-Gowers theorem

- X. Shao
- Mathematicsdiscrete Analysis
- 26 November 2018

We deduce, as a consequence of the arithmetic removal lemma, an almost-all version of the Balog-Szemer\'{e}di-Gowers theorem: For any $K\geq 1$ and $\varepsilon > 0$, there exists $\delta =… Expand

## Type-II/III DCT/DST algorithms with reduced number of arithmetic operations

- X. Shao, S. Johnson
- Computer ScienceSignal Process.
- 29 March 2007

## A robust version of Freiman's 3k–4 Theorem and applications

Abstract We prove a robust version of Freiman's 3k – 4 theorem on the restricted sumset A+ΓB, which applies when the doubling constant is at most (3+$\sqrt{5}$)/2 in general and at most 3 in the… Expand

## A Density Version of the Vinogradov Three Primes Theorem

- X. Shao
- Mathematics
- 26 June 2012

We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 5/8, then all sufficiently large odd positive integers can be written as the sum of three primes… Expand

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