Neil Lyall
It is a striking and elegant fact (proved independently by Furstenberg and Sarkozy) that in any subset of the natural numbers of positive upper density there necessarily exist two distinct elements whose difference is given by a perfect square. In this article we present a new and simple proof of this result by adapting an argument originally developed by Croot and Sisask to give a new proof of Roth's theorem.
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We extend the best known bound on the largest subset of {1,2,...,N} with no square differences to the largest possible class of quadratic polynomials.
We define a necessary and sufficient condition on a polynomial $h\in \mathbb{Z}[x]$ to guarantee that every set of natural numbers of positive upper density contains a nonzero difference of the form $h(p)$ for some prime $p$. Moreover, we establish a quantitative estimate on the size of the largest subset of ${1,2,\dots,N}$ which lacks the desired arithmetic structure, showing that if deg$(h)=k$, then the density of such a set is at most a constant times $(\log N)^{-c}$ for any $c<1/(2k-2)$. We also discuss how an improved version of this result for $k=2$ and a relative version in the primes c…
Sárközy proved that dense sets of integers contain two elements differing by a $k$th power. The bounds in quantitative versions of this theorem are rather weak compared to what is expected. We prove a version of Sárközy's theorem for polynomials over $\mathbb{F}_q$ with polynomial dependencies in the parameters. More precisely, let $P_{q,n}$ be the space of polynomials over $\mathbb{F}_q$ of degree $< n$ in an indeterminate $T$. Let $k \geq 2$ be an integer and let $q$ be a prime power. Set $c(k,q) := (2 k^2 D_q(k)^2\log q)^{-1}$, where $D_q(k)$ is the sum of the digits of $k$ in base $q$. If …
Using Fourier analytic techniques, we prove that if $\VE>0$, $N\geq \exp\exp(C\VE^{-1}\log\VE^{-1})$ and $A\subseteq\{1,...,N\}$, then there must exist $t\in\N$ such that \[\frac{|A\cap (A+t^2)|}{N}>(\frac{|A|}{N})^2-\VE.\] This is a special case of results presented in Lyall and Magyar \cite{LM3} and we will follow those arguments closely. We hope that the exposition of this special case will serve to illuminate the key ideas contained in \cite{LM3}, where many of the analogous arguments are significantly more technical.
New bounds for the Furstenberg-Sárközy theorem
本文改进 Furstenberg–Sárközy 定理，证明无平方差整数集的大小满足新的指数衰减上界。
Furstenberg--Sárközy theorem over number fields
本文在数域整数环上引入相交多项式与上密度概念，证明正上密度子集必含两元素差为某多项式值，并给出定量版本，推广了Furstenberg--Sárközy定理。
Sárközy's Theorem in Various Finite Field Settings
本文加强了Green在多项式环设定下关于Sarkozy定理的研究，去除了多项式根数与$q$互质的限制条件，并在有限域$\mathbb{F}_{p^n}$中获得了具有类似强界的定理版本。
We show that if $h\in \mathbb{Z}[x]$ is a polynomial of degree $k \geq 2$ such that $h(\mathbb{N})$ contains a multiple of $q$ for every $q\in \mathbb{N}$, known as an $\textit{intersective polynomial}$, then any subset of $\{1,2,\dots,N\}$ with no nonzero differences of the form $h(n)$ for $n\in\mathbb{N}$ has density at most a constant depending on $h$ and $c$ times $(\log N)^{-c\log\log\log\log N}$, for any $c<(\log((k^2+k)/2))^{-1}$. Bounds of this type were previously known only for monomials and intersective quadratics, and this is currently the best-known bound for the original Furstenb…
The Furstenberg-Sárközy theorem for polynomials in one or more prime variables
本文研究集合中缺失由素数变量多项式生成的非零差的最大子集大小，证明了与无限制整数输入情形已知最佳上界同类型的结果。
Extensions of the Furstenberg-Sárközy theorem via the arithmetic level-$d$ inequality
本文基于Green和Sawhney的方法，通过算术水平-d不等式，为不含非零交集多项式差的集合提供了一个准多项式上界，改进了密度增量方案。
Polynomial actions of rings of integers of global fields and quasirandomness of Paley-type graphs
本文在整体域整数环框架下给出 Furstenberg–Sárközy 定理的充要条件，并将其推广到有限主理想环，进而利用渐近完全遍历性与拟随机性的双向联系构造新的代数拟随机图族。
The Furstenberg-Sárközy theorem for sums of an even number of odd powers
本文证明了差集不含奇数幂偶数项和的集合满足幂次节省上界，并通过贪心构造说明该界形状最优。
Furstenberg--Sárközy theorem and partition regularity of polynomial equations over finite fields
研究有限域上多项式配置的组合性质，改进Furstenberg--Sárközy定理的界限，并证明多项式方程的分拆正则性。
We generalize the Sarkozy-Furstenberg theorem on squares in difference sets of integers, and show that, given any positive definite function f:Z_N->C with density at least r(N), where r(N)=O((\log N)^{-c}), there is a perfect square s<=N/2 such that f(s) is non-zero. We do not rely on the usual analysis of the dichotomy of randomness and periodicity of a set and iterative application of the Hardy-Littlewood method. Instead, we find a bound for the van der Corput property of the set of squares.
We present a short new proof of the canonical polynomial van der Waerden theorem, recently established by Girao [arXiv:2004.07766].
An uncountable number of proofs of Pythagoras Theorem
本研究提出了无穷多个毕达哥拉斯定理的证明方法，其中部分证明可归类为自相似证明，拓展了该定理的证明思路。
We give here a new proof of a Tauberian Theorem of complex Laplace transform using the Theory of measure and theory of function with bounded variations. However we deduce the simple proof of Prime Number Theorem.
In this paper, we give a detailed account of Goldfeld's proof of Siegel's theorem. Particularly, we present complete proofs of the nontrivial assumptions made in his paper.
We give a new proof of Tietze Theorem on the convergence of infinite semi-regular continued fractions.
We present a proof of a multidimensional version of Peres-Schlag's theorem on Diophantine approximations with lacunary sequences.
We present a proof of Moessner's theorem by double induction, using only basic rules of arithmetic. No prerequisite knowledge is assumed. Familiarity with summation is advised.
We give an analytic proof of the norm index theorem $[I_:K^* N(I_L)] =[L:K]$ for cyclic extensions of number fields using spectral theory of the idele class group.
We present a new, elementary, dynamical proof of the prime number theorem.
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