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Suppose that $k\geq 2$ and $A$ is a non-empty subset of a finite abelian group $G$ with $|G|>1$. Then the cardinality of the restricted sumset $$ k^\wedge A:=\{a_1+\cdots+a_k:\,a_1,\ldots,a_k\in A,\ a_i\neq a_j\text{ for }i\neq j\} $$ is at least $$ \min\{p(G), k|A|-k^2+1\}, $$ where $p(G)$ denotes the least prime divisor of $|G|$., Comment: 9 pages. Accepted by the Israel Journal of Mathematics

Optical tweezers can manipulate tiny particles. However, the distortion caused by the scattering medium restricts the applications of optical tweezers. Wavefront shaping techniques including the transmission matrix (TM) method are powerful tools to achieve light focusing behind the scattering medium. In this paper, we propose a new kind of TM, named intensity transmission matrix (ITM). Only relying on the intensity distribution, we can calculate the ITM with only about 1/4 measurement time of the widely used four-phase method. Meanwhile, ITM method can avoid the energy loss in diffraction introduced by holographic modulation. Based on the ITM, we have implemented particle manipulation with a high degree of freedom on single and multiple particles. In addition, the manipulation range is enlarged over twenty times (compared with the memory effect) to 200 {\mu}m., Comment: 21 pages, 5 figures

We prove that, for any $c_1,c_2,c_3\in(1,41/35)$, every sufficiently large odd number $N$ can be represented as the sum of three primes $N = p_1 + p_2 +p_3$ such that $p_i = \lfloor n_{i}^{c_i}\rfloor$ for some $n_i \in{\mathbb N}$ for each $1 \leq i \leq 3$. Our arguments are based on a variant of Green's transference principle due to Matom\"aki, Maynard and Shao. We prove a necessary restriction estimate using Bourgain's strategy and employ Harman's sieve method to optimize our upper bound for $c_i$., Comment: 42 pages

Let $q$ be an odd prime power. Combining the discussion of Varnavides and a recent theorem of Ellenberg and Gijswijt, we show that a subset $A\subset{\mathbb F}_q^n$ will contain many non-trivial three-term arithmetic progressions, whenever $|A|\geq (c_q q)^n$ for some constant $c_q>0$. After the first version of our manuscript was uploaded in the arXiv, we learned from Professors Jacob Fox and Terence Tao that our result is a special case of a result of Fox and Lovasz [1, Theorem 3]. In fact, [1, Theorem 3] gives a much better bound than ours. For example, when $q=3$, the lower bound given by Fox and Lovasz is $|A|^{2}\cdot (|A|q^{-n})^{11.901}$, while our bound is $|A|^{2}\cdot (|A|q^{-n})^{25.803}$. We thank Professors Jacob Fox and Terence Tao for their helpful comments on our manuscript. [1] Jacob Fox, L\'aszl\'o Mikl\'os Lov\'asz, A tight bound for Green's arithmetic triangle removal lemma in vector spaces, preprint, arXiv:1606.01230., Comment: Our result is weaker than a known result of Fox and Lovasz

Suppose that $A$, $B$ and $S$ are non-empty subsets of a finite abelian group $G$. Then the generalized restricted sumset $$ A\stackrel{S}+B:=\{a+b:\,a\in A,\ b\in B,\ a-b\not\in S\} $$ contains at least $$ \min\{|A|+|B|-3|S|,p(G)\} $$ elements, where $p(G)$ is the least prime factor of $|G|$. Further, we also have $$ |A\stackrel{S}+B|\geq \min\{|A|+|B|-|S|-2,p(G)\}, $$ provided that both $|A|$ and $|B|$ are large with respect to $|S|$., Comment: This is a very preliminary draft

We prove a generalization of Frieman's $3k-3$ theorem for the sumset $$ \Sigma^{l}(A_1,\ldots,A_k)=\{a_{j_{1}}+\cdots+a_{j_{l}}:\,1\leq j_{1}<\cdots

Suppose that $A,B$ are two non-empty subsets of the finite nilpotent group $G$. If $A\not=B$, then the cardinality of the restricted sumset $$A\dotplus B={a+b: a\in A, b\in B, a\neq b} $$ is at least $$\min{p(G),|A|+|B|-2},$$ where $p(G)$ denotes the least prime factor of $|G|$., Comment: This is a preliminary draft, which maybe contains some mistakes. Now Theorem 1.2 has been extended to general finite groups