Thomas F. Bloom, James Maynard
We show that if $A\subset \{1,\ldots,N\}$ has no solutions to $a-b=n^2$ with $a,b\in A$ and $n\geq 1$ then \[|A|\ll \frac{N}{(\log N)^{c\log\log \log N}}\] for some absolute constant $c>0$. This improves upon a result of Pintz-Steiger-Szemerédi.
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Let $S_2$ be the set of integer squares. We show that the dimension $d$ of a Hilbert cube $a_0+\{0,a_1\}+...+ \{0, a_d\}\subset S_2$ is bounded by $d=O(\log \log N)$
We prove that if $\varepsilon(m)\to 0$ arbitrarily slowly, then for almost all $m$ and any $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero quadratic residues we have $|A|\leq m^{1/2-\varepsilon(m)}.$
We prove that if $A\subseteq \{ 1,2,\dots, N \}$ does not contain any solution to the equation $x_1+\dots+x_k=y_1+\dots+y_k$ with distinct $x_1,\dots,x_k,y_1,\dots,y_k\in A$, then $|A|\ll {k^{3/2}}N^{1/k}.$
For infinitely many primes $p=4k+1$ we give a slightly improved upper bound for the maximal cardinality of a set $B\subset \ZZ_p$ such that the difference set $B-B$ contains only quadratic residues. Namely, instead of the "trivial" bound $|B|\leq \sqrt{p}$ we prove $|B|\leq \sqrt{p}-1$, under suitable conditions on $p$. The new bound is valid for approximately three quarters of the primes $p=4k+1$.
For $g \geq 2$ and $h \geq 3$, we give small improvements on the maximum size of a $B_h[g]$-set contained in the interval $\{1,2, \dots , N \}$. In particular, we show that a $B_3[g]$-set in $\{1,2, \dots , N \}$ has at most $(14.3 g N)^{1/3}$ elements. The previously best known bound was $(16 gN)^{1/3}$ proved by Cilleruelo, Ruzsa, and Trujillo. We also introduce a related optimization problem that may be of independent interest.
We present a sharp extension of a result of Bourgain on finding configurations of $k+1$ points in general position in measurable subset of $\mathbb{R}^d$ of positive upper density whenever $d\geq k+1$ to all proper $k$-degenerate distance graphs.
We say that a set is a multiplicative 3-Sidon set if the equation $s_1s_2s_3=t_1t_2t_3$ does not have a solution consisting of distinct elements taken from this set. In this paper we show that the size of a multiplicative 3-Sidon subset of $\{1,2,\dots,n\}$ is at most $π(n)+π(n/2)+n^{2/3}(\log n )^{2^{1/3}-1/3+o(1)}$, which improves the previously known best bound $π(n)+π(n/2)+cn^{2/3}\log n/\log\log n$.
Let $G$ be an abelian group. A set $A \subset G$ is a \emph{$B_k^+$-set} if whenever $a_1 + \dots + a_k = b_1 + \dots + b_k$ with $a_i, b_j \in A$ there is an $i$ and a $j$ such that $a_i = b_j$. If $A$ is a $B_k$-set then it is also a $B_k^+$-set but the converse is not true in general. Determining the largest size of a $B_k$-set in the interval $\{1, 2, \dots, N \} \subset \integers$ or in the cyclic group $\integers_N$ is a well studied problem. In this paper we investigate the corresponding problem for $B_k^+$-sets. We prove non-trivial upper bounds on the maximum size of a $B_k^+$-set con…
The problem of looking for subsets of the natural numbers which contain no 3-term arithmetic progressions has a rich history. Roth's theorem famously shows that any such subset cannot have positive upper density. In contrast, Rankin in 1960 suggested looking at subsets without three-term geometric progressions, and constructed such a subset with density about 0.719. More recently, several authors have found upper bounds for the upper density of such sets. We significantly improve upon these bounds, and demonstrate a method of constructing sets with a greater upper density than Rankin's set. Th…
A generalization of Croot-Lev-Pach's Lemma and a new upper bound for the size of difference sets in polynomial rings
本文推广Croot-Lev-Pach多项式引理，结合陶哲轩切片秩估计与Gröbner基技术，改进Green的结果，给出多项式环中差集大小及无等差数列集合大小的新上界。
In this entry point into the subject, combining two elementary proofs, we decrease the gap between the upper and lower bounds by $0.2\%$ in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of $\{ 1, 2, \ldots, n\}$ is at most $\sqrt{n}+ 0.998n^{1/4}$ for sufficiently large $n$.
We present a strengthening of the lemma on the lower bound of the slice rank by Tao (2016) motivated by the Croot-Lev-Pach-Ellenberg-Gijswijt bound on cap sets (2017, 2017). The Croot-Lev-Pach-Ellenberg-Gijswijt method and the lemma of Tao are based on the fact that the rank of a diagonal matrix is equal to the number of non-zero diagonal entries. Our lemma is based on the rank of upper-triangular matrices. This stronger lemma allows us to prove the following extension of the Ellenberg-Gijswijt result (2017). A tricolored ordered sum-free set in $\mathbb F_p^n$ is a collection $\{(a_i,b_i,c_i)…
Based on work of P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, we show that if a covering system has distinct squarefree moduli, then the minimum modulus is at most 118. We also show that in general the $k^{\rm th}$ smallest modulus in a covering system with distinct moduli (provided it is required for the covering) is bounded by an absolute constant.
Sieving with square conditions and applications to Hilbert cubes in arithmetic sets
本文探讨了Gallagher筛法在模素数平方下的应用局限，并提出通过转换余数信息以提升筛法效果，同时研究了Hilbert立方体在不同乘法定义集中的最大维度，得出了相关上界。
We discuss the problem of finding distinct integer sets $\{x_1,x_2,...,x_n\}$ where each sum $x_i+x_j, i \ne j$ is a square, and $n \le 7$. We confirm minimal results of Lagrange and Nicolas for $n=5$ and for the related problem with triples. We provide new solution sets for $n=6$ to add to the single known set. This provides new information for problem D15 in Guy's {\it Unsolved Problems in Number Theory}
A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets "greedily," a generalization of…
Starting from a result of Stewart, Tijdeman and Ruzsa on iterated difference sequences, we introduce the notion of iterated compositions of linear operations. We prove a general result on the stability of such compositions (with bounded coefficients) on sets of integers having a positive upper density.
We define a random Liouville function (λ_Q) which depends on a random set (Q) of primes and prove that (A_Q = \{n \in \mathbb{N} | λ_Q(n) = -1 \}) is normal almost everywhere. This fact enables us to generate a family of normal sets such that the equation (xy =z) is not solvable inside them. Additionally we prove that equations (xy=z^2, x^2 + y^2 = square, x^2 - y^2 = square) are solvable in any normal set and for any equation (xy=cn^2) ((c > 1 ), is not a square) there exists a normal set (A_c) such that the equation is not solvable inside (A_c).
Let $R_s(n)$ denote the number of representations of the positive number $n$ as the sum of two squares and $s$ biquadrates. When $s=3$ or $4$, it is established that the anticipated asymptotic formula for $R_s(n)$ holds for all $n\le X$ with at most $O(X^{(9-2s)/8+\varepsilon})$ exceptions.
Abstract upper densities are monotone and subadditive functions from the power set of positive integers to the unit real interval that generalize the upper densities used in number theory, including the upper asymptotic density, the upper Banach density, and the upper logarithmic density. We answer a question posed by G. Grekos in 2013, and prove the existence of translation invariant abstract upper densities onto the unit interval, whose null sets are precisely the family of finite sets, or the family of sequences whose series of reciprocals converge. We also show that no such density can be …
Every set $A$ of positive integers with upper Banach density 1 contains an infinite sequence of pairwise disjoint subsets $(B_i)_{i=1}^{\infty}$ such that $B_i$ has upper Banach density 1 for all $i \in \mathbf{N}$ and $\sum_{i\in I} B_i \subseteq A$ for every nonempty finite set $I$ of positive integers.
Differences of squares of upper-triangular $2\times 2$ integer matrices
研究上三角整数矩阵能表示为两个上三角整数矩阵平方差的条件，给出基于数的平方差表示和额外整除条件的完整判据，并分类可表示矩阵的同余条件。
We find the exact size of a maximal non-commuting set in unipotent uppertriangular linear group $UU_4(\mathbb{F}_q)$ in terms of a non-commuting geometric structure (Refer Definition [10]), where $\mathbb{F}_q$ is the finite field with $q$ elements. Then we get bounds on the size of such a set by explicitly finding certain non-commuting sets in the non-commuting structure.
On the upper bound of wavefront sets of representations of p-adic groups
本文研究 p-adic 连通约化群不可约容许表示的波前集上界，提出新猜想并将其归结为反离散系列表示，进而证明它与局部 Arthur 包及 ABV 包中的相应猜想等价。
Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a satisfactory way. Their natural generalization, called B 2 [g] sets and defined by the fact that there are at most g ways (up to reordering the summands) to represent a given integer as a sum of two elements of the set, are much more difficult to handle and not as well understood. In this article, using a numerical approach, we improve the best upper estimates …
Let $P$ and $T$ be disjoint sets of prime numbers with $T$ finite. A simple formula is given for the natural density of the set of square-free numbers which are divisible by all of the primes in $T$ and by none of the primes in $P$. If $P$ is the set of primes congruent to $r$ modulo $m$ (where $m$ and $r$ are relatively prime numbers), then this natural density is shown to be $0$.
Squares, three fleas, sporadic sets, and squares
该研究探讨平面上三只跳蚤按规则跳跃形成的三角形面积，发现所有正半整数可实现，缺失整数为完全平方数及5、29、80、99、179，揭示其与阿波罗尼奥斯群相关的几何结构及平方数障碍。
Explicit Upper Bounds on Decay Rates of Fourier Transforms of Self-similar Measures on Self-similar Sets
本文研究自相似集上自相似概率测度的傅里叶变换，导出了衰减率的显式上界以改进先前研究，并应用于Lüroth表示数字受限的数集。
We prove a conjecture of the first-named author ([J14]) on the upper bound Fourier coefficients of automorphic forms in Arthur packets of all classical groups over any number field. This conjecture generalizes the global version of the local tempered $L$-packet conjecture of F. Shahidi ([Sh90] and [Sh10]).
We present a general technique for obtaining permutation polynomials over a finite field from permutations of a subfield. By applying this technique to the simplest classes of permutation polynomials on the subfield, we obtain several new families of permutation polynomials. Some of these have the additional property that both f(x) and f(x)+x induce permutations of the field, which has combinatorial consequences. We use some of our permutation polynomials to exhibit complete sets of mutually orthogonal latin squares. In addition, we solve the open problem from a recent paper by Wu and Lin, and…
It is proved that \[ \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N\exp\left(C\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right) \] holds for arbitrary integers $1\le n_1<\cdots < n_N$. This bound is essentially better than that found in a recent paper of Aistleitner, Berkes, and Seip and can not be improved by more than possibly a power of $1/\sqrt{\log\log\log N}$. The proof relies on ideas from classical work of Gál, the method of Aistleitner, Berkes, and Seip, and a certain completeness property of extremal sets of square-free numbers.
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。
We prove upper bounds for the mean square of the remainder in the prime geodesic theorem, for every cofinite Fuchsian group, which improve on average on the best known pointwise bounds. The proof relies on the Selberg trace formula. For the modular group we prove a refined upper bound by using the Kuznetsov trace formula.
Cardinalities of Jordan Diophantine sets of upper triangular integer matrices
该研究确定了上三角整数矩阵环中Jordan D(N)-集存在无限集的N的条件，明确了有限集的最大基数，还给出了两类特殊无限集存在的充要条件。
We consider the linear vector space formed by the elements of the finite fields $\mathbb{F}_q$ with $q=p^r$ over $\mathbb{F}_p$. Let $\{a_1,\ldots,a_r\}$ be a basis of this space. Then the elements $x$ of $\mathbb{F}_q$ have a unique representation in the form $\sum_{j=1}^r c_ja_j$ with $c_j\in\mathbb{F}_p$. Let $D_1,\ldots,D_r$ be subsets of $\mathbb{F}_p$. We consider the set $W=W(D_1,\ldots,D_r)$ of elements of $\mathbb{F}_q$ such that $c_j \in D_j$ for all $j=1,\ldots,r$. We give an estimate for the number of squares in the set $W$ which implies an asymptotic formula for this quantity in t…
We show that if $h\in\mathbb{Z}[x]$ is a polynomial of degree $k$ such that the congruence $h(x)\equiv0\pmod{q}$ has a solution for every positive integer $q$, then any subset of $\{1,2,\ldots,N\}$ with no two distinct elements with difference of the form $h(n)$, with $n$ positive integer, has density at most $(\log N)^{-c\log\log\log N}$, for some constant $c$ that depends only on $k$. This improves on the best bound in the literature, due to Rice, and generalizes a recent result of Bloom and Maynard.
A prime plus a square: A conditional 11/20 upper bound
本文在本原Dirichlet L函数非平凡零点实部≤7/8的假设下，结合导体加权均方估计等方法，得到不能表示为素数加平方数的正整数计数的11/20上界。
We prove a lower and an upper bound for the large sieve with square moduli for function fields. These bounds correspond to bounds for the classical large sieve with square moduli established in arXiv:1812.05844 by Baier, Lynch and Zhao and arXiv:math/0512271 by Baier and Zhao. Our lower bound in the function field setting contradicts an upper bound obtained in arXiv:1802.03131 by Baier and Singh. Indeed, we point out an error in arXiv:1802.03131.
On square-free numbers generated from given sets of primes II
该研究延续此前工作，分析由指定素数集生成的不超过x的正无平方因子数计数函数Q_𝒫(x)的渐近行为，重点探究素数集满足Dirichlet特征χ(p)=1条件下的函数性质。
On square-free numbers generated from given sets of primes
本文研究由给定素数集生成的无平方因子数的计数问题，在素数集大小随参数单调增长且趋于无穷的条件下，分析不超过某实数的这类整数的数量估计。
On simultaneous approximation to a real number, its square, and its cube, II
本文解释了此前关于几何级数中四个$\mathbb{Q}$-线性无关实数的一致有理逼近指数上界为何不是最优的。
Let f be a square-free polynomial in Fq[t][x] where Fq is a field of q elements. We view f as a polynomial in the variable x with coefficients in the ring Fq[t]. We study squarefree values of f in sparse subsets of Fq[t] which are given by a linear condition. The motivation for our study is an analogue problem of representing square-free integers by integer polynomials, where it is conjectured that setting aside some simple exceptional cases, a square-free polynomial f in Z[x] takes infinitely many square-free values. Let c(t) be a polynomial in Fq[t] of degree less than m, and let k < m be co…
We give an upper bound for the exponential sum over squarefree integers. This establishes a conjecture by Brüdern and Perelli.
Topology of zero sets of polynomials with square discriminant
本研究针对满足特定封闭性的整数系数集合，证明了该集合系数的整系数多项式零点可由同集合系数、平方判别式的整系数多项式零点任意逼近，揭示这类多项式零点集复闭包的拓扑结构与判别式是否为平方无关，区别于伽罗瓦群的相关性质。
Delone sets are locally finite point sets, such that (a) any two points are separated by a given minimum distance, and (b) there is a given radius so that every ball of that radius contains at least one point. Important examples include the vertex set of Penrose tilings and other regular model sets, which serve as a mathematical model for quasicrystals. In this note we show that the point set given by the values $\sqrt{n} e^{2πi α\sqrt{n}}$ with $n=1,2,3,\ldots$ is a Delone set in the complex plane, for any $α>0$. This complements Akiyama's recent observation (see arXiv:1904.10815) that $\sqrt…
In this paper we obtain new lower bounds for the upper box dimension of $αβ$ sets. As a corollary of our main result, we show that if $α$ is not a Liouville number and $β$ is a Liouville number, then the upper box dimension of any $αβ$ set is $1$. We also use our dimension bounds to obtain new results on affine embeddings of self-similar sets.
An upper quasi-density on $\bf H$ (the integers or the non-negative integers) is a real-valued subadditive function $μ^\ast$ defined on the whole power set of $\mathbf H$ such that $μ^\ast(X) \le μ^\ast({\bf H}) = 1$ and $μ^\ast(k \cdot X + h) = \frac{1}{k}\, μ^\ast(X)$ for all $X \subseteq \bf H$, $k \in {\bf N}^+$, and $h \in \bf N$, where $k \cdot X := \{kx: x \in X\}$; and an upper density on $\bf H$ is an upper quasi-density on $\bf H$ that is non-decreasing with respect to inclusion. We say that a set $X \subseteq \bf H$ is small if $μ^\ast(X) = 0$ for every upper quasi-density $μ^\ast$ …
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