Let $f_{s,k}(n)$ be the maximum possible number of $s$-term arithmetic progressions in a sequence $a_1<a_2<\ldots<a_n$ of $n$ integers which contains no $k$-term arithmetic progression. For all integers $k > s \geq 3$, we prove that $$\lim_{n \to \infty} \frac{\log f_{s,k}(n)}{\log n} = 2,$$ which answers an old question of Erdős. In fact, we prove upper and lower bounds for $f_{s,k}(n)$ which show that its growth is closely related to the bounds in Szemerédi's theorem.
We construct subsets of {1,...,N} of cardinality at least N exp(-C(log N)^{1/(k+1)}) which do not contain arithmetic progressions of length 2^k+1. This extends a result of Behrend (1946) concerning sets which do not contain aritmetic progressions of length 3.
We present an elementary proof that if $A$ is a finite set of numbers, and the sumset $A+_GA$ is small, $|A+_GA|\leq c|A|$, along a dense graph $G$, then $A$ contains $k$-term arithmetic progressions.
We show that for integer $n>0$, any subset $A \subset Z_4^n$ free of three-term arithmetic progressions has size $|A| < 4^{c n}$, with an absolute constant $c \approx 0.926$.
In this paper we prove: If 0 < d < 1, and p is a sufficiently large prime, then if S is a subset of Z/pZ having the least number of three-term arithmetic progressions among all subsets of Z/pZ having at least dp elements, then S has an arithmetic progression of length at least log^{1/4+o(1)} x.
We prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of prime numbers. The proof uses the probabilistic method. Using the same techniques we improve the bounds obtained by He for gaps in geometric-progression-free sets.
In order to investigate multiplicative structures in additively large sets, Beiglböck et al. raised a significant open question as to whether or not every subset of the natural numbers with bounded gaps (syndetic set) contains arbitrarily long geometric progressions. A result of Erdős implies that syndetic sets contain a $2$-term geometric progression with integer common ratio, but we still do not know if they contain such a progression with common ratio being a perfect square. In this article, we prove that for each $k\in \mathbb{N}$, a syndetic set contains $2$-term geometric progressions wi…
We consider the problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length $k$ in a given set of size $n$. It is proved that it is sufficient, in a certain sense, to consider the interval $[1,\dots, n]$. The study continues the work of Komlós, Sulyok, and Szemerédi.
This paper is mainly concerned with sets which do not contain four-term arithmetic progressions, but are still very rich in three term arithmetic progressions, in the sense that all sufficiently large subsets contain at least one such progression. We prove that there exists a positive constant $c$ and a set $A \subset \mathbb F_q^n$ which does not contain a four-term arithmetic progression, with the property that for every subset $A' \subset A$ with $|A'| \geq |A|^{1-c}$, $A'$ contains a nontrivial three term arithmetic progression. We derive this from a more general quantitative Roth-type the…
We show that $\sqrt{k}\cdot r^{k/2}$ is a threshold interval length where, under mild conditions, almost every $r$-coloring of an interval of longer length contains a monochromatic $k$-term arithmetic progression, while almost no $r$-coloring of an interval of shorter length contains a monochromatic $k$-term arithmetic progression.
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 $d=O(\log \log N)$
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…
We show that sets avoiding 6-term arithmetic progressions in $\mathbb{Z}_6^n$ have size at most $5.709^n$. It is also pointed out that the "product construction" does not work in this setting, specially, we show that for the extremal sizes in small dimensions we have $r_6(\mathbb{Z}_6)=5$, $r_6(\mathbb{Z}_6^2)=25$ and $ 116\leq r_6(\mathbb{Z}_6^3)\leq 124$.
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…
We show that if a finite, large enough subset A of an arbitrary abelian group satisfies the small doubling condition |A + A| < (log |A|)^{1 - epsilon} |A|, then A must contain a three-term arithmetic progression whose terms are not all equal, and A + A must contain an arithmetic progression or a coset of a subgroup, either of which of size at least exp^[ c (log |A|)^{delta} ]. This extends analogous results obtained by Sanders and, respectively, by Croot, Laba and Sisask in the case where the group is that of the integers or a finite field.
A conjecture of Freiman gives an exact formula for the largest volume of a finite set $A$ of integers with given cardinality $k = |A|$ and doubling $T = |2A|$. The formula is known to hold when $T \le 3k-4$, for some small range over $3k-4$ and for families of structured sets called chains. In this paper we extend the formula to sets of every dimension and prove it for sets composed of three segments, giving structural results for the extremal case. A weaker extension to sets composed of a bounded number of segments is also discussed.
Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be polynomials with distinct degrees, each having zero constant term. We show that any subset $A$ of $\{1,\dots,N\}$ with no nontrivial progressions of the form $x,x+P_1(y),\dots,x+P_m(y)$ has size $|A|\ll N/(\log\log{N})^{c_{P_1,\dots,P_m}}$. Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.
Let $A, B\subseteq \mathbb{Z}$ be finite, nonempty subsets with $\min A=\min B=0$, and let $$δ(A,B)={\begin{array}{ll} 1 & \hbox{if} A\subseteq B, 0 & \hbox{otherwise.} If $\max B\leq \max A\leq |A|+|B|-3$ and \label{one}|A+B|\leq |A|+2|B|-3-δ(A,B), then we show $A+B$ contains an arithmetic progression with difference 1 and length $|A|+|B|-1$. As a corollary, if \eqref{one} holds, $\max(B)\leq \max(A)$ and either $\gcd(A)=1$ or else $\gcd(A+B)=1$ and $|A+B|\leq 2|A|+|B|-3$, then $A+B$ contains an arithmetic progression with difference 1 and length $|A|+|B|-1$.
Furstenberg, Glasscock, Bergelson, Beiglboeck have been studied abundance in arithmatic progression on various large sets like piecewise syndetic, central, thick, etc. but also there are so many sets in which abundance in progression is still unsettled like J-sets, C-sets, D-sets etc. But all of these sets have a common property that they contains arbitrary length of arithmatic progressions. These type of sets are called sets of A.P. rich, we have given an elementary proof of abundance of those sets.
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.
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.
Fix a prime p and a density 0 < d <= 1. Among all functions f : F_p -> [0,1], what can one say about those which assign minimal weight to three-term arithmetic progressions -- that is, the sum of f(a)f(a+x)f(a+2x) is minimal as we sum over all a and x -- subject to the density constraint that the expected value of f equals d? In the present paper we show three things about them: 1) Such f are nearly indicator functions; 2) They enjoy a certain ``local minimal'' property; and, 3) They are approximately indicator functions for certain sumsets A+B.
We construct Salem sets in $\mathbb{R}/\mathbb{Z}$ of any dimension (including $1$) which do not contain any arithmetic progressions of length $3$. Moreover, the sets can be taken to be Ahlfors regular if the dimension is less than $1$, and the measure witnessing the Fourier decay can be taken to be Frostman in the case of dimension $1$. This is in sharp contrast to the situation in the discrete setting (where Fourier uniformity is well known to imply existence of progressions), and helps clarify a result of Laba and Pramanik on pseudo-random subsets of the real line which do contain progressi…
Assuming the well-known conjecture that [x,x+x^t] contains a prime for t > 0 and x sufficiently large, we prove: For 0 < r < 1, there exists 0 < s < r < 1, 0 < d < 1, and infinitely many primes q such that if S is a subset of Z/qZ having density at least s, and having the least number of 3-term arithemtic progressions among all sets of density at least s, then S is nearly translation invariant in a very strong sense. Namely, there exists 0 <= b <= q-1 such that |S intersect (S + bj)| = (1-g(s))|S|, for every 0 < j < q^d, where g(s) -> 0 as s -> 0. A curious feature of the proof is that Behrend…
We study progression-free sets in the abelian groups $G=(\mathbb{Z}_m^n,+)$. Let $r_k(\mathbb{Z}_m^n)$ denote the maximal size of a set $S \subset \mathbb{Z}_m^n$ that does not contain a proper arithmetic progression of length $k$. We give lower bound constructions, which e.g. include that $r_3(\mathbb{Z}_m^n) \geq C_m \frac{((m+2)/2)^n}{\sqrt{n}}$, when $m$ is even. When $m=4$ this is of order at least $3^n/\sqrt{n}\gg \vert G \vert^{0.7924}$. Moreover, if the progression-free set $S\subset \mathbb{Z}_4^n$ satisfies a technical condition, which dominates the problem at least in low dimension,…
For an integer $b \geqslant 2$ and a set $S\subset \{0,\cdots,b-1\}$, we define the Kempner set $\mathcal{K}(S,b)$ to be the set of all non-negative integers whose base-$b$ digital expansions contain only digits from $S$. These well-studied sparse sets provide a rich setting for additive number theory, and in this paper we study various questions relating to the appearance of arithmetic progressions in these sets. In particular, for all $b$ we determine exactly the maximal length of an arithmetic progression that omits a base-$b$ digit.
We consider two problems regarding arithmetic progressions in symmetric sets in the finite field (product space) model. First, we show that a symmetric set $S\subseteq\mathbb{Z}_q^n$ containing $|S|=μ\cdot q^n$ elements must contain at least $δ(q,μ)\cdot q^n\cdot 2^n$ arithmetic progressions $x,x+d,\ldots,x+(q-1)\cdot d$ such that the difference $d$ is restricted to lie in $\{0,1\}^n$. Second, we show that for prime $p$ a symmetric set $S\subseteq\mathbb{F}^n_p$ with $|S|=μ\cdot p^n$ elements contains at least $μ^{C(p)}\cdot p^{2n}$ arithmetic progressions of length $p$. This establishes that …
Pach and Palincza proved the following generalization of Ellenberg and Gijswijt's bound for the size of $k$-term arithmetic progression-free subsets, where $k\in \{4,5,6\}$: Let $m>0$ be an integer such that $6$ divides $m$ and let $k\in \{4,5,6\}$. Then $$ r_k({\mathbb Z}_{m}^n)\leq (0.948m)^n $$ if $n$ is sufficiently large. Building on the proof technique of Pach and Palincza's upper bound we generalize the Ellenberg and Gijswijt's bound in the following way: Let $p>2$ be any integer and let $q>2$ be a prime. Suppose that $p\leq q$. Then the there exists an $n_0\in \mathbb N$ integer and a …
Let $G$ be a finite group, and let $r_{3}(G)$ represent the size of the largest subset of $G$ without non-trivial three-term progressions. In a recent breakthrough, Croot, Lev and Pach proved that $r_{3}(C_{4}^{n}) \leqslant (3.61)^{n}$, where $C_{m}$ denotes the cyclic group of order $m$. For finite abelian groups $G \cong \prod_{i=1}^{n} C_{m_{i}}$, where $m_{1},\ldots,m_{n}$ denote positive integers such that $m_{1} | \ldots | m_{n}$, this also yields a bound of the form $r_{3}(G) \leqslant (0.903)^{\operatorname{rk}_{4}(G)} |G|$, with $\operatorname{rk}_{4}(G)$ representing the number of i…
Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise syndetic set contains k length arithmetic progression and such collection is also piecewise syndetic in Z: They used algebraic structure of beta N. The above result was extended for arbitrary semigroups by Bergelson and Hindman, again using the structure of Stone-Cech compactification of general semigroup. However they provided the abundances for various types of large sets. But the abundances in in many large sets is still unknown. In this work we will provide the abundance in Quasi Central sets, J-sets and C-sets.
Alternating geometric progressions modulo one and Sturmian words
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心）；University of Chinese Academy of Sciences（中国科学院大学）
We show that any subset of $\mathbb{Z}_p^n$ ($p$ an odd prime) without $3$-term arithmetic progression has size $O(p^{cn})$, where $c:=1-\frac{1}{18\log p}<1$. In particular, we find an upper bound of $O(2.84^n)$ on the maximum size of an affine cap in $GF(3)^n$.
A celebrated and deep result of Green and Tao states that the primes contain arbitrarily long arithmetic progressions. In this note I provide a straightforward argument demonstrating that the primes get arbitrarily close to arbitrarily long arithmetic progressions. The argument also applies to `large sets' in the sense of Erdős-Turán. The proof is short, completely self-contained, and aims to give a heuristic explanation of why the primes, and other large sets, possess arithmetic structure.
Additive combinatorics is built around the famous theorem by Szemerédi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemerédi's theorem is an existence statement, whereas the ultimate goal in combinatorics is always to make enumeration statements. In this article we develop new methods based on real algebraic geometry to obtain several quantitative statements on the number of arithmetic progressions in fixed density sets. We further discuss the possibility of a genera…
Using a new graphical representation for partitions, the author obtains a family of partition identities associated with partitions into distinct parts of an arithmetic progression, or, more generally, with partitions into distinct parts of a set that is a finite union of arithmetic progressions associated with a modular sum-free Sidon set. Partition identities are also constructed for sets associated with modular sum-free sets.
Motzkin posed the problem of finding the maximal density $μ(M)$ of sets of integers in which the differences given by a set $M$ do not occur. The problem is already settled when $|M|\leq 2$ and $M$ is a finite arithmetic progression. In this paper, we determine $μ(M)$ when $M$ has some other structure. For example, we determine $μ(M)$ when $M$ is a finite geometric progression.
A set of positive integers $A \subset \mathbb{Z}_{> 0}$ is \emph{log-sparse} if there is an absolute constant $C$ so that for any positive integer $x$ the sequence contains at most $C$ elements in the interval $[x,2x)$. In this note we study arithmetic progressions in sums of log-sparse subsets of $\mathbb{Z}_{> 0}$. We prove that for any log-sparse subsets $S_1, \dots, S_n$ of $\mathbb{Z}_{> 0},$ the sumset $S = S_1 + \cdots + S_n$ cannot contain an arithmetic progression of size greater than $n^{(1+o(1))n}.$ We also show that this is nearly tight by proving that there exist log-sparse sets $…
A $\textit{ladder}$ is a set $S \subseteq \mathbb Z^+$ such that any finite coloring of $\mathbb Z$ contains arbitrarily long monochromatic progressions with common difference in $S$. Van der Waerden's theorem famously asserts that $\mathbb Z^+$ itself is a ladder. We also discuss variants of ladders, namely $\textit{accessible}$ and $\textit{walkable}$ sets, which are sets $S$ such that any coloring of $\mathbb Z$ contains arbitrarily long (for accessible sets) or infinite (for walkable sets) monochromatic sequences with consecutive differences in $S$. We show that sets with upper density 1 a…
We provide estimates for the dimensions of sets in $\mathbb{R}$ which uniformly avoid finite arithmetic progressions. More precisely, we say $F$ uniformly avoids arithmetic progressions of length $k \geq 3$ if there is an $ε>0$ such that one cannot find an arithmetic progression of length $k$ and gap length $Δ>0$ inside the $εΔ$ neighbourhood of $F$. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of $k$ and $ε$. In the other direction, we provide examples of sets which uniformly avoid arithmetic progressions of a given length but…
Given two sets $\cA, \cB \subseteq \F_q$ of elements of the finite field $\F_q$ of $q$ elements, we show that the productset $$ \cA\cB = \{ab | a \in \cA, b \in\cB\} $$ contains an arithmetic progression of length $k \ge 3$ provided that $k<p$, where $p$ is the characteristic of $\F_q$, and $# \cA # \cB \ge 3q^{2d-2/k}$. We also consider geometric progressions in a shifted productset $\cA\cB +h$, for $f \in \F_q$, and obtain a similar result.
Impartial subtraction games on the nonnegative integers have been studied by many and discussed in detail in for example the remarkable work Winning Ways by Conway, Berlekamp and Guy. We describe how comply variations of these games, similar to those introduced by Holshouser, Reiter, Smith, Stănică, can be defined as having its sets of winning positions identical to well-known sets avoiding arithmetic progressions such as $x+z=2y$, studied by Szerkeres, Erd\H os and Turán, and many others, thus exploring a new territory combining ideas from combinatorial games and combinatorial number theory. …
Ellenberg and Gijswijt gave recently a new exponential upper bound for the size of three-term arithmetic progression free sets in $({\mathbb Z_p})^n$, where $p$ is a prime. Petrov summarized their method and generalized their result to linear forms. In this short note we use Petrov's result to give new exponential upper bounds for the Erdős-Ginzburg-Ziv constant of finite Abelian groups of high rank. Our main results depend on a conjecture about Property D.
For integers $m$ and $n$, we study the problem of finding good lower bounds for the size of progression-free sets in $(\mathbb{Z}_{m}^{n},+)$. Let $r_{k}(\mathbb{Z}_{m}^{n})$ denote the maximal size of a subset of $\mathbb{Z}_{m}^{n}$ without arithmetic progressions of length $k$ and let $P^{-}(m)$ denote the least prime factor of $m$. We construct explicit progression-free sets and obtain the following improved lower bounds for $r_{k}(\mathbb{Z}_{m}^{n})$: If $k\geq 5$ is odd and $P^{-}(m)\geq (k+2)/2$, then \[r_k(\mathbb{Z}_m^n) \gg_{m,k} \frac{\bigl\lfloor \frac{k-1}{k+1}m +1\bigr\rfloor^{n…
We provide quantitative estimates for the supremum of the Hausdorff dimension of sets in the real line which avoid $\varepsilon$-approximations of arithmetic progressions. Some of these estimates are in terms of Szemerédi bounds. In particular, we answer a question of Fraser, Saito and Yu (IMRN, 2019) and considerably improve their bounds. We also show that Hausdorff dimension is equivalent to box or Assouad dimension for this problem, and obtain a lower bound for Fourier dimension.
Green and Sisask showed that the maximal number of $3$-term arithmetic progressions in $n$-element sets of integers is $\lceil n^2/2\rceil$; it is easy to see that the same holds if the set of integers is replaced by the real line or by any Euclidean space. We study this problem in general metric spaces, where a triple $(a,b,c)$ of points in a metric space is considered a $3$-term arithmetic progression if $d(a,b)=d(b,c)=\frac{1}{2}d(a,c)$. In particular, we show that the result of Green and Sisask extends to any Cartan--Hadamard manifold (in particular, to the hyperbolic spaces), but does not…
