I. D. Shkredov
Let A \subseteq [1,..,N]^2 be a set of cardinality at least N^2/(log log N)^c, where c>0 is an absolute constant. We prove that A contains a triple {(k,m), (k+d,m), (k,m+d)}, where d>0. This theorem is a two-dimensional generalization of Szemeredi's theorem on arithmetic progression.
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Using the density-increment strategy of Roth and Gowers, we derive Szemeredi's theorem on arithmetic progressions from the inverse conjectures GI(s) for the Gowers norms, recently established by the authors and Ziegler.
Let G be a finite Abelian group and A be a subset G\times G of cardinality at least |G|^2/(log log |G|)^c, where c>0 is an absolute constant. We prove that A contains a triple {(k,m), (k+d,m), (k,m+d)}, where d does not equal 0. This theorem is a two-dimensional generalization of Szemeredi's theorem on arithmetic progressions.
We consider, over both the integers and finite fields, Szemerédi's theorem on $k$-term arithmetic progressions where the set $S$ of allowed common differences in those progressions is restricted and random. Fleshing out a line of enquiry suggested by Frantzikinakis et al, we show that over the integers, the conjectured threshold for $\mathbb{P}(d \in S)$ for Szemerédi's theorem to hold a.a.s follows from a conjecture about how so-called dual functions are approximated by nilsequences. We also show that the threshold over finite fields is different to this threshold over the integers.
Note on the Theorem of Balog, Szemerédi, and Gowers
该研究针对Balog-Szemerédi-Gowers定理的结构化集合规模问题，证明满足特定能量条件的加性集存在更大规模的子集，且其差集大小有对应上界，给出了该定理下能得到的最大结构化集合结果。
We deduce, as a consequence of the arithmetic removal lemma, an almost-all version of the Balog-Szemerédi-Gowers theorem: For any $K\geq 1$ and $\varepsilon > 0$, there exists $δ= δ(K,\varepsilon)>0$ such that the following statement holds: if $|A+_ΓA| \leq K|A|$ for some $Γ\geq (1-δ)|A|^2$, then there is a subset $A' \subset A$ with $|A'| \geq (1-\varepsilon)|A|$ such that $|A'+A'| \leq |A+_ΓA| + \varepsilon |A|$. We also discuss issues around quantitative bounds in this statement, in particular showing that when $A \subset \mathbb{Z}$ the dependence of $δ$ on $ε$ cannot be polynomial for any…
Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be any linearly independent polynomials with zero constant term. We show that there exists a $γ>0$ such that any subset of $\mathbb{F}_q$ of size at least $q^{1-γ}$ contains a nontrivial polynomial progression $x,x+P_1(y),\dots,x+P_m(y)$, provided the characteristic of $\mathbb{F}_q$ is large enough.
We obtain quantitative versions of the Balog-Szemeredi-Gowers and Freiman theorems in the model case of a finite field geometry F_2^n, improving the previously known bounds in such theorems. For instance, if A is a subset of F_2^n such that |A+A| <= K|A| (thus A has small additive doubling), we show that there exists an affine subspace V of F_2^n of cardinality |V| >> K^{-O(\sqrt{K})} |A| such that |A \cap V| >> |V|/2K. Under the assumption that A contains at least |A|^3/K quadruples with a_1 + a_2 + a_3 + a_4 = 0 we obtain a similar result, albeit with the slightly weaker condition |V| >> K^{…
Uniform nonlinear Szemerédi theorem for corners in finite fields
本文针对有限域平面子集，证明了由有理函数生成的非线性角构型计数的渐近公式，推广了角的 Szemerédi 型结论。
Improved Bounds for Szemerédi's Theorem
本文利用Gowers $U^k$范数逆定理的拟多项式界和密度增量策略，改进了$k\ge5$时无$k$项等差数列集合的上界。
Further bounds in the polynomial Szemerédi theorem
该研究针对多项式Szemerédi定理，证明满足特定大小条件的整数子集包含指定非平凡模式，是除等差数列细化外首个获整数定量界的高复杂度构型。
Let F be a fixed finite field of characteristic at least 5. Let G = F^n be the n-dimensional vector space over F, and write N := |G|. We show that if A is a subset of G with size at least c_F N(log N)^{-c}, for some absolute constant c > 0 and some c_F > 0, then A contains four distinct elements in arithmetic progression. This is equivalent, in the usual notation of additive combinatorics, to the assertion that r_4(G) <<_F N(log N)^{-c}.
Let p > 4 be a prime. We show that the largest subset of F_p^n with no 4-term arithmetic progressions has cardinality << N(log N)^{-c}, where c = 2^{-22} and N := p^n. A result of this type was claimed in a previous paper by the authors and published in Proc. London Math. Society. Unfortunately the proof had a gap, and we issue an erratum for that paper here. Our new argument is different and significantly shorter. In fact we prove a stronger result, which can be viewed as a quantatitive version of some previous results of Bergelson-Host-Kra and the authors.
Let $r_k(n)$ denote the maximum cardinality of a set $A \subset \{1,2, \dots, n \}$ such that $A$ does not contain a $k$-term arithmetic progression. In this paper, we give a method of constructing such a set and prove the lower bound $n^{1-\frac{c_k}{k \ln k}} < r_k(n)$ where $k$ is prime, and $c_k \rightarrow 1$ as $k \rightarrow \infty$. This bound is the best known for an increasingly large interval of $n$ as we choose larger and larger $k$. We also demonstrate that one can prove or disprove a conjecture of Erdős on arithmetic progressions in large sets once tight enough bounds on $r_k(n)$…
We construct large subsets of the first $N$ positive integers which avoid certain arithmetic configurations. In particular, we construct a set of order $N^{0.7685}$ lacking the configuration $\{x,x+y,x+y^2\},$ surpassing the $N^{3/4}$ limit of Ruzsa's construction for sets lacking a square difference. We also extend Ruzsa's construction to sets lacking polynomial differences for a wide class of univariate polynomials. Finally, we turn to multivariate differences, constructing a set of order $N^{1/2}$ lacking a difference equal to a sum of two squares. This is in contrast to the analogous probl…
Szemerédi's Theorem Along Cantor Sets of Integers
该研究在整数Cantor集上扩展了Szemerédi定理，证明了正上Banach密度子集包含特定等差数列的结论。
A Multidimensional Szemerédi Theorem in the primes
本文证明素数多维组中任意有限集均存在仿射副本，推广了Green-Tao定理，其通过非均匀超图伪随机权重系统并扩展超图移除引理证明完成。
We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi's theorem, which asserts that any subset of the integers of positive density contains progressions of arbitrary length. The second, which is the main new ingredient of this paper, is a certain transference principle. This allows us to deduce from Szemeredi's theorem that any subset of a sufficiently pseudorandom set of positive relative density contains progressions of arbitrary length. The third ingredient is a recent result of Goldston and Yildirim. Using this…
We prove a quantitative version of the Polynomial Szemeredi Theorem for difference sets. This result is achieved by first establishing a higher dimensional analogue of a theorem of Sarkozy (the simplest non-trivial case of the Polynomial Szemeredi Theorem asserting that the difference set of any subset of the integers of positive upper density necessarily contains a perfect square) and then applying a simple lifting argument.
New bounds for Szemeredi's theorem, II: A new bound for $r_4(N)$
本文改进了不含四项等差数列集合最大基数$r_4(N)$的上界，将Gowers 1998年的结果提升至$r_4(N) \ll N e^{-c\sqrt{\log \log N}}$，并预告第三部分将作进一步改进。
A Quantitative Bound For Szemerédi's Theorem for a Complexity One Polynomial Progression over $\mathbb{Z}/N\mathbb{Z}$
本文针对 $\mathbb{Z}/N\mathbb{Z}$ 上一类复杂度一的四项多项式级数，改造 Peluse 的次数降低方法并结合二次 Fourier 分析，给出缺失该级数集合大小的迭代对数定量界。
We extend a recent result of Khalfalah and Szemeredi to the polynomials of prime variables.
An old conjecture of Graham stated that if $n$ is a prime and $S$ is a sequence of $n$ terms from the cyclic group $C_n$ such that all (nontrivial) zero-sum subsequences have the same length, then $S$ must contain at most two distinct terms. In 1976, Erdős and Szemeredi gave a proof of the conjecture for sufficiently large primes $n$. However, the proof was complicated enough that the details for small primes were never worked out. Both in the paper of Erdős and Szemeredi and in a later survey by Erdős and Graham, the complexity of the proof was lamented. Recently, a new proof, valid even for …
On polynomial progressions via transference
本文利用迁移原理，在素数模群和整数情形下建立多项式 Szemerédi 定理的新合理界，并给出避免固定多项式公差模式集合的定量上界。
We show that there exists $c>0$ such that any subset of $\{1, \dots, N\}$ of density at least $(\log\log{N})^{-c}$ contains a nontrivial progression of the form $x,x+y,x+y^2$. This is the first quantitatively effective version of the Bergelson--Leibman polynomial Szemerédi theorem for a progression involving polynomials of differing degrees. Our key innovation is an inverse theorem characterising sets for which the number of configurations $x,x+y,x+y^2$ deviates substantially from the expected value. In proving this, we develop the first effective instance of a concatenation theorem of Tao and…
In an additive group (G,+), a three-dimensional corner is the four points g, g+d(1,0,0), g+d(0,1,0), g+d(0,0,1), where g is in G^3, and d is a non-zero element of G. The Ramsey number of interest is R_3(G) the maximal cardinality of a subset of G^3 that does not contain a three-dimensional corner. Furstenberg and Katznelson have shown R_3(Z_N) is little-o of N^3, and in fact the corresponding result holds in all dimensions, a result that is a far reaching extension of the Szemeredi Theorem. We give a new proof of the finite field version of this fact, a proof that is a common generalization of…
We prove a generalisation of Roth's theorem for arithmetic progressions to d-configurations, which are sets of the form {n_i+n_j+a}_{1 \leq i \leq j \leq d} where a, n_1,..., n_d are nonnegative integers, using Roth's original density increment strategy and Gowers uniformity norms. Then we use this generalisation to improve a result of Sudakov, Szemerédi and Vu about sum-free subsets and prove that any set of n integers contains a sum-free subset of size at least log n (log log log n)^{1/32772 - o(1)}.
Let $AP_k=\{a,a+d,\ldots,a+(k-1)d\}$ be an arithmetic progression. For $ε>0$ we call a set $AP_k(ε)=\{x_0,\ldots,x_{k-1}\}$ an $ε$-approximate arithmetic progression if for some $a$ and $d$, $|x_i-(a+id)|<εd$ holds for all $i\in\{0,1\ldots,k-1\}$. Complementing earlier results of Dumitrescu, in this paper we study numerical aspects of Van der Waerden, Szemeredi and Furstenberg-Katznelson like results in which arithmetic progressions and their higher dimensional extensions are replaced by their $ε$-approximation.
We show that under the assumption of a 24-term version of Fermat's Last Theorem, there exists an absolute constant c > 0 such that if S is a set of n > n_0 positive integers satisfying |S.S| < n^(1+c), then the sumset S.S satisfies |S+S| >> n^2. In other words, we prove a weak form of the Erdos-Szemeredi sum-product conjecture, conditional on an extension of Fermat's Last Theorem. Unconditionally, we prove this theorem for when S is a set of n monic polynomials. We also prove an analogue of a theorem of Bourgain and Chang for the ring C[x].
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