On sumsets containing a perfect square
英文摘要：
We show $A+B$ contains a perfect square if $A,B \subseteq \{1,\dots,N\}$ have $|A|,|B| \ge (\frac{3}{8}+ε)N$. The constant $\frac{3}{8}$ is optimal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
相关论文
Sumsets in primes containing almost all even positive integers
Let $A$ be a subset of primes up to $x$. If we assume $A$ is well-distributed (in the Siegel-Walfisz sense) in any arithmetic progressions to moduli $q\leqslant(\log x)^c$ for any $c>0$, then the sumset $A+A$ has density 1/2 in the natural numbers as $x$ tends to infinity, which also yields almost all even positive integers could be represented as the sums of two primes in $A$ as $x$ tends to infinity. This result, improving the previous results in such special case, could be compared with the classical estimation for the exceptional set of binary Goldbach problem.
math.COmath.NT
乘法子群中的受限和集
Restricted sumsets in multiplicative subgroups
Georgia Institute of Technology（佐治亚理工学院）
本文研究了有限域上乘法子群及整数完美幂中的受限和集，扩展了Shkredov的结果，并提出了一个受限和集的van Lint-MacWilliams猜想的类比。
math.COmath.NT
The additive structure of the squares inside rings
When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the set's underlying structure. We begin by investigating finite sets of perfect squares and associated sumsets. We reveal how arithmetic progressions efficiently reduce the cardinality of sumsets and provide estimates for the minimum size, taking advantage of the additive structure that arithmetic progressions provide. We then generalise the problem to arbitrary rings and achieve satisfactory estimates for the case of squares in…
math.COmath.NT
使得两个集合的和集中的所有元素均为完全平方数的两个整数集合
Two sets of integers such that all elements of the sumset of the two sets are perfect squares
本文研究寻找两个整数集合使其所有两两之和均为完全平方数的问题，在 $(m,n)$ 为 $(3,3)$、$(5,3)$ 和 $(4,4)$ 三种情况下获得了多个多参数解，并指出了获取更多解的方法。
math.NT
Three-term arithmetic progressions and sumsets
Suppose that G is an abelian group and A is a finite subset of G containing no three-term arithmetic progressions. We show that |A+A| >> |A|(log |A|)^{1/3-ε} for all ε>0.
math.CAmath.NT
关于和集的密度，II
On the density of sumsets, II
该研究针对算术拟密度下和集的密度问题，证明对于覆盖模n!的o(n!)个剩余类的非空集B，存在集A使A与A+B在所有算术拟密度定义域内且和集密度可取[0,1]中任意值，拓展了和集密度理论。
math.NT
更多相关论文收起更多论文
关于具有小和集的集合数量
On the number of sets with small sumset
本文研究阿贝尔群中具有小和集的子集数量，通过改进的容器型定理给出精确上界，并证明随机小和集几乎包含在小算术级数中，从而改进 Campos 等人的结果并几乎解决 Alon 等人的猜想。
math.COmath.NT
Large gaps between consecutive prime numbers containing square-free numbers and perfect powers of prime numbers
We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers.
math.NT
Large gaps between consecutive prime numbers containing perfect powers
For any positive integer $k$, we show that infinitely often, perfect $k$-th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size $$ c_k \frac{\log p \log_2 p \log_4 p}{(\log_3 p)^2}, $$ where $p$ is the smaller of the two primes.
math.NT
Large gaps between consecutive prime numbers containing perfect $k$-th powers of prime numbers
Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a fixed positive constant, for which the interval $(p_n, p_{n+1})$ contains the $k$-th power of a prime number.
math.NT
Strongly perfect lattices sandwiched between Barnes-Wall lattices
New series of $2^{2m}$-dimensional universally strongly perfect lattices $Λ_I $ and $Γ_J $ are constructed with $$2BW_{2m} ^{\#} \subseteq Γ_J \subseteq BW_{2m} \subseteq Λ_I \subseteq BW _{2m}^{\#} .$$ The lattices are found by restricting the spin representations of the automorphism group of the Barnes-Wall lattice to its subgroup ${\mathcal U}_m:={\mathcal C}_m (4^H_{\bf 1}) $. The group ${\mathcal U}_m$ is the Clifford-Weil group associated to the Hermitian self-dual codes over ${\bf F} _4$ containing ${\bf 1}$, so the ring of polynomial invariants of ${\mathcal U}_m$ is spanned by the gen…
math.COmath.NTmath.RT
包含强对的有理丢番图六元组
Rational Diophantine sextuples with strong pair
本文研究了有理丢番图多元组问题，通过引入强对条件，成功构造出包含强丢番图对的有理丢番图六元组的无限族。
math.NT
