Tom Sanders
We show that if A is a subset of F_2^n and |A+A| < K|A| then A is contained in a subspace of size at most 2^{O(K^{3/2}log K)}|A|. This improves on the previous best of 2^{O(K^2)}.
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A famous result of Freiman describes the structure of finite sets A of integers with small doubling property. If |A + A| <= K|A| then A is contained within a multidimensional arithmetic progression of dimension d(K) and size f(K)|A|. Here we prove an analogous statement valid for subsets of an arbitrary abelian group.
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^{…
We prove a robust version of Freiman's $3k - 4$ theorem on the restricted sumset $A+_ΓB$, which applies when the doubling constant is at most $\tfrac{3+\sqrt{5}}{2}$ in general and at most $3$ in the special case when $A = -B$. As applications, we derive robust results with other types of assumptions on popular sums, and structure theorems for sets satisfying almost equalities in discrete and continuous versions of the Riesz-Sobolev inequality.
On Freiman's Theorem in a function field setting
本研究在函数域框架下证明了推广Freiman $3k-4$定理的乘法版猜想的新实例，验证了代数闭域上有理函数域中满足特定维数条件的向量空间的猜想成立。
We note a link between combinatorial results of Bollobás and Leader concerning sumsets in the grid, the Brunn-Minkowski theorem and a result of Freiman and Bilu concerning the structure of sets of integers with small doubling. Our main result is the following. If eps > 0 and if A is a finite nonempty subset of a torsion-free abelian group with |A + A| <= K|A|, then A may be covered by exp(K^C) progressions of dimension [log_2 K + eps] and size at most |A|.
Green and Ruzsa recently proved that for any $s\ge2$, any small squaring set $A$ in a (multiplicative) abelian group, i.e. $|A\cdot A|<K|A|$, has a Freiman $s$-model: it means that there exists a group $G$ and a Freiman $s$-isomorphism from $A$ into $G$ such that $|G|<f(s,K)|A|$. In an unpublished note, Green proved that such a result does not necessarily hold in non abelian groups if $s\ge64$. The aim of this paper is improve Green's result by showing that it remains true under the weaker assumption $s\ge6$.
Freiman's 2.4-Theorem states that any set $A \subset \mathbb{Z}_p$ satisfying $|2A| \leq 2.4|A| - 3 $ and $|A| < p/35$ can be covered by an arithmetic progression of length at most $|2A| - |A| + 1$. A more general result of Green and Ruzsa implies that this covering property holds for any set satisfying $|2A| \leq 3|A| - 4$ as long as the rather strong density requirement $|A| < p/10^{215}$ is satisfied. We present a version of this statement that allows for sets satisfying $|2A| \leq 2.48|A| - 7$ with the more modest density requirement of $|A| < p/10^{10}$.
We discuss a multiplicative counterpart of Freiman's $3k-4$ theorem in the context of a function field $F$ over an algebraically closed field $K$. Such a theorem would give a precise description of subspaces $S$, such that the space $S^2$ spanned by products of elements of $S$ satisfies $\dim S^2 \leq 3 \dim S-4$. We make a step in this direction by giving a complete characterisation of spaces $S$ such that $\dim S^2 = 2 \dim S$. We show that, up to multiplication by a constant field element, such a space $S$ is included in a function field of genus $0$ or $1$. In particular if the genus is $1…
Using various results from extremal set theory (interpreted in the language of additive combinatorics), we prove an asyptotically sharp version of Freiman's theorem in F_2^n: if A in F_2^n is a set for which |A + A| <= K|A| then A is contained in a subspace of size 2^{2K + O(\sqrt{K}\log K)}|A|; except for the O(\sqrt{K} \log K) error, this is best possible. If in addition we assume that A is a downset, then we can also cover A by O(K^{46}) translates of a coordinate subspace of size at most |A|, thereby verifying the so-called polynomial Freiman-Ruzsa conjecture in this case. A common theme i…
Characterisation of Meyer sets via the Freiman--Ruzsa theorem
本研究利用刻画有界翻倍有限集的Freiman--Ruzsa定理，给出了欧氏空间中差集一致离散的相对稠密子集即Meyer集刻画的新证明方法。
We show that if A is a large subset of a box in Z^d with dimensions L_1 >= L_2 >= ... >= L_d which are all reasonably large, then |A + A| > 2^{d/48}|A|. By combining this with Chang's quantitative version of Freiman's theorem, we prove a structural result about sets with small sumset. If A is a set of integers with |A + A| <= K|A|, then there is a progression P of dimension d << log K such that |A \cap P| >= \exp(-K^C)max (|A|, |P|). This is closely related to a theorem of Freiman and Bilu, but is quantitatively stronger in certain aspects. [Added Oct 14th: I have temporarily withdrawn this pa…
The Polynomial Freiman-Ruzsa conjecture is one of the central open problems in additive combinatorics. If true, it would give tight quantitative bounds relating combinatorial and algebraic notions of approximate subgroups. In this note, we restrict our attention to subsets of Euclidean space. In this regime, the original conjecture considers approximate algebraic subgroups as the set of lattice points in a convex body. Green asked in 2007 whether this can be simplified to a generalized arithmetic progression, while not losing more than a polynomial factor in the underlying parameters. We give …
Freiman's $3k-4$ Theorem for Function Fields
本文将整数情形的 Freiman $3k-4$ 定理推广到函数域，证明当扩张中 $k$ 维向量空间 $S$ 的两两乘积生成空间的维数不超过 $3k-4$ 时，$K(S)$ 必为小亏格函数域，且 $S$ 在某个 Riemann-Roch 空间中具有小余维数。
We prove a Freiman-type theorem for locally compact abelian groups. If A is a subset of a locally compact abelian group with Haar measure m and m(nA) < n^dm(A) for all n>d log d then we describe A in a way which is tight up to logarithmic factors in d.
Improved Bounds for the Freiman-Ruzsa Theorem
本文改进了弗雷米安-鲁兹萨定理的界，通过熵方法和傅里叶分析相结合，证明了在特定条件下有限子集可被有限个凸余集进展覆盖，并接近多项式猜想的结论。
We show that if $A$ is a finite $K$-approximate subgroup of an $s$-step nilpotent group then there is a finite normal subgroup $H\subset A^{K^{O_s(1)}}$ modulo which $A^{O_s(\log^{O_s(1)}K)}$ contains a nilprogression of rank at most $O_s(\log^{O_s(1)}K)$ and size at least $\exp(-O_s(\log^{O_s(1)}K))|A|$. This partially generalises the close-to-optimal bounds obtained in the abelian case by Sanders, and improves the bounds and simplifies the exposition of an earlier result of the author. Combined with results of Breuillard-Green, Breuillard-Green-Tao, Gill-Helfgott-Pyber-Szabó, and the author,…
A quasipolynomial inverse theorem for the $\mathsf{U}^k(\mathbb{F}_p^n)$ norm in the high characteristic
本文证明了高特征下 Gowers 范数的拟多项式逆定理，通过建立 Freiman 多同态的结构定理，结合代数正则化、BSG 定理和多线性映射理论，显著简化了证明。
Let A be a finite set of integers. We prove that if |A| is at least 2 and |A+A| is 3|A|-3, then one of the following is true: 1. A is a bi-arithmetic progression; 2. A+A contains an arithmetic progression of length 2|A|-1; 3. |A| is 6 and A is Freiman isomorphic to the set {(0,0),(0,1),(0,2),(1,0),(1,1),(2,0)}; 4. A is Freiman isomorphic to a set in either the form of {0,2,...,2k} union B union {n} for some non-negative integer k at most n/2 -2 or the form of {0} union C union D union {n}, where n=2|A|-2, B is left dense in [2k,n-1], C is right dense in [1,u] for some u in [4,n-6], D is left d…
Geometric lifting and Freiman's $3k-4$ theorem in compact connected abelian groups
本文提出几何提升方法，在紧致连通阿贝尔群中证明Freiman $3k-4$定理的类比，并建立最优投影定理，解决Christ-Iliopoulou问题，推广至流行和集与Tao卷积不等式。
A polynomial Freiman-Ruzsa inverse theorem for function fields
本文利用 Gowers 等人的最新证明，在函数域上证明了多项式 Freiman-Ruzsa 猜想，并给出了有限集加法大小的最优下界。
For $δ>0$ sufficiently small and $A\subset \mathbb{Z}^k$ with $|A+A|\le (2^k+δ)|A|$, we show either $A$ is covered by $m_k(δ)$ parallel hyperplanes, or satisfies $|\widehat{\operatorname{co}}(A)\setminus A|\le c_kδ|A|$, where $\widehat{\operatorname{co}}(A)$ is the smallest convex progression (convex set intersected with a sublattice) containing $A$. This generalizes the Freiman-Bilu $2^k$ theorem, Freiman's $3|A|-4$ theorem, and recent sharp stability results of the present authors for sumsets in $\mathbb{R}^k$ conjectured by Figalli and Jerison.
On Waring's problem: beyond Freiman's theorem
本文将 Freiman 关于 Waring 问题的定理有效化，证明了指数倒数和的具体下界条件，并在指数序列构成等差数列时得出更强的整数表示结论。
The well--known Freiman--Ruzsa Theorem provides a structural description of a set $A$ of integers with $|2A|\le c|A|$ as a subset of a $d$--dimensional arithmetic progression $P$ with $|P|\le c'|A|$, where $d$ and $c'$ depend only on $c$. The estimation of the constants $d$ and $c'$ involved in the statement has been the object of intense research. Freiman conjectured in 2008 a formula for the largest volume of such a set. In this paper we prove the conjecture for a general class of sets called chains.
New bounds in the Bogolyubov-Ruzsa lemma
该研究建立了Bogolyubov-Ruzsa引理的新界，给出有限阿贝尔群中密度为α的子集A对应的3A-3A所含Bohr集的秩与半径界，推进了多项式Freiman-Ruzsa猜想研究并改进了Freiman定理的界。
Freiman's $(3k-4)$-like results for subset and subsequence sums
本文受 Freiman $3k-4$ 定理启发，利用 $h$ 重和集与限制和集表示，证明了子集和与子序列和集合的相关结构定理，并推广至广义形式，同时为 Nathanson 等人的已有结果提供了新证明。
A note on transverse sets and bilinear varieties
本文通过直接组合方法证明了密集横截集包含有界余维数的双线性流形，改进了界并避免了傅里叶分析和弗里曼定理及其变种。
We show two results. First, a refinement of Freiman's theorem: if A is a finite set of integers and |A+A| < K|A|, then A is contained in a multidimensional progression of dimension at most O(K^{7/4} log^3K) and size at most exp(O(K^{7/4} log^3K))|A|. Secondly, an improvement of a result of Konyagin and Laba: if A is a finite set of reals and a is a transcendental then |A+aA| >> |A|(log |A|)^{4/3-ε} for all ε>0.
Additively stable sets, critical sets for the 3k-4 theorem in $\mathbb{Z}$ and $\mathbb{R}$
本文研究整数与实数中的加法左稳定集，给出其在初始区间内密度的精确上界及达到该界的构造，并用于揭示 Freiman $3k-4$ 定理临界集的结构。
We survey the classical results on the prime number theorem
In this note, we investigate J.-C. Liu's work on truncated Gauss' square exponent theorem and obtain more truncations. We also discuss some possible multiple summation extensions of Liu's results.
A note on Kronecker's approximation theorem
本文针对 Kronecker 逼近定理，改进了 Gonek-Montgomery 给出的定量表述。
Hardy's theorem for the Riemann zeta-function $ζ(s)$ says that it admits infinitely many complex zeros on the line $\Re({s}) = \frac{1}{2}$. In this note, we give a simple proof of this statement which, to the best of our knowledge, is new.
A note on the theorem of Maynard and Tao
本文基于梅纳德和陶的定理，以更强形式重新证明了Shiu关于连续同余素数'字符串'的结果。
In this note, we improve earlier results towards the Bruinier-Kohnen sign equidistribution conjecture for half-integral weight modular eigenforms in terms of natural density by using a consequence of Halász' Theorem. Moreover, applying a result of Serre we remove all unproved assumptions.
Given a prime p and a positive integer m satisfying a certain inequality, the converse of Wolstenholme's Theorem is shown to hold for the product mp^k where k is any positive integer, generalizing a result by Helou and Terjanian.
We generalise a result of Heath-Brown and Skorobogatov to show that a certain class of varieties over a number field $k$ satisfies Weak Approximation and the Hasse Principle, provided there is no Brauer-Manin obstruction.
In the setting of the integers, Granville, Harper and Soundararajan showed that the upper bound in Halász's Theorem can be improved for smoothly supported functions. We derive the analogous result for Halász's Theorem in $\mathbb{F}_q[t]$, and then consider the converse question of when the general upper bound in this version of Halász's Theorem is actually attained.
We survey the classical results of the Dirichlet Approximation Theorem.
In this note, we apply Moriwaki's arithmetic height functions to obtain an analogue of Silverman's Specialization Theorem for families of Abelian varieties over $K$, where $K$ is any field finitely generated over ${\Q}$.
In this short note we present an elementary proof of the Ax-Lindemann-Weierstrass theorem for abelian and semi-abelian varieties. The proof uses ideas of Pila, Ullmo, Yafaev, Zannier and is based on basic properties of sets definable in o-minimal structures. It does not use the Pila-Wilkie counting theorem.
A Note on the Phragmen-Lindelof Theorem
本文对弗拉格门-林德勒夫定理进行了推广，旨在纠正文献中的错误。
We extend a result of Greenberg and Stevens on the interpolation of modular symbols in Hida families to the context of non-split rational quaternion algebras. Both the definite case and the indefinite case are considered.
We show that it is impossible to get a better remainder than the classical one in the Wiener-Ikehara theorem even if one assumes analytic continuation of the Mellin transform after subtraction of the pole to a half-plane. We also prove a similar result for the Ingham-Karamata theorem.
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