Three-term Arithmetic Progression Research Articles

Characterization of Order Structures Avoiding Three-term Arithmetic Progressions

Characterization of Order Structures Avoiding Three-term Arithmetic Progressions

The Kelley–Meka bounds for sets free of three-term arithmetic progressions

The Kelley–Meka bounds for sets free of three-term arithmetic progressions

Roth's theorem and the Hardy–Littlewood majorant problem for thin subsets of primes

We introduce a wide class of deterministic subsets of primes of zero relative density and we prove Roth's Theorem in these sets, namely, we show that any subset of them with positive relative upper density contains infinitely many non-trivial three-term arithmetic progressions. We also prove that the Hardy–Littlewood majorant property holds for these subsets of primes. Notably, our considerations recover the results for the Piatetski–Shapiro primes for exponents close to 1, which are primes of the form ⌊nc⌋ for a fixed c>1.

Three-term arithmetic progressions, including at least two primes of the intervals [n2, (n+1)2

Three-term arithmetic progressions, including at least two primes of the intervals [n2, (n+1)2

Tower-type bounds for Roth’s theorem with popular differences

Green developed an arithmetic regularity lemma to prove a strengthening of Roth's theorem on arithmetic progressions in dense sets. It states that for every $\epsilon > 0$ there is some $N\_0(\epsilon)$ such that for every $N \ge N\_0(\epsilon)$ and $A \subset \[N]$ with $|A| = \alpha N$, there is some nonzero $d$ such that $A$ contains at least $(\alpha^3 - \epsilon) N$ three-term arithmetic progressions with common difference $d$. We prove that the minimum $N\_0(\epsilon)$ in Green's theorem is an exponential tower of twos of height on the order of $\log(1/\epsilon)$. Both the lower and upper bounds are new. This shows that the tower-type bounds that arise from the use of a regularity lemma in this application are quantitatively necessary.

Popular Differences for Right Isosceles Triangles

For a subset $A$ of $\{1,2,\ldots,N\}^2$ of size $\alpha N^2$ we show existence of $(m,n)\neq(0,0)$ such that the set $A$ contains at least $(\alpha^3 - o(1))N^2$ triples of points of the form $(a,b)$, $(a+m,b+n)$, $(a-n,b+m)$. This answers a question by Ackelsberg, Bergelson, and Best. The same approach also establishes the corresponding result for compact abelian groups. Furthermore, for a finite field $\mathbb{F}_q$ we comment on exponential smallness of subsets of $(\mathbb{F}_q^n)^2$ that avoid the aforementioned configuration. The proofs are minor modifications of the existing proofs regarding three-term arithmetic progressions.

Improved bound in Roth's theorem on arithmetic progressions

We prove that if A⊆{1,…,N} does not contain any non-trivial three-term arithmetic progression, then|A|≪(log⁡log⁡N)3+o(1)log⁡NN.

Boxes, extended boxes and sets of positive upper density in the Euclidean space

AbstractWe prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

Three-term arithmetic progressions in subsets of F_q^∞ of large Fourier dimension

We show that subsets of \(\mathbf{F}_q^{\infty}\) of large Fourier dimension must contain three-term arithmetic progressions. This contrasts with a construction of Shmerkin of a subset of \(\mathbf{R}\) of Fourier dimension 1 with no three-term arithmetic progressions.

Maximum subsets of $${{\mathbb {F}}}^n_q$$ containing no right angles

Recently, Croot, Lev and Pach (Ann Math 185:331–337, 2017) and Ellenberg and Gijswijt (Ann Math 185:339–443, 2017) developed a new polynomial method and used it to prove upper bounds for three-term arithmetic progression free sets in \({\mathbb {Z}}_4^n\) and \({\mathbb {F}}_3^n\), respectively. Their approach was later summarized by Tao and is now known as the slice rank method. In this paper, we apply this method to obtain a new upper bound on the cardinality of subsets of \({\mathbb {F}}^n_q\) which contain no right angles. More precisely, let q be a fixed odd prime power and \(x\cdot y\) be the standard inner product of two vectors \(x,y\in {\mathbb {F}}_q^n\), and we prove that the maximum cardinality of a subset \(A\subseteq {\mathbb {F}}_q^n\) without three distinct elements \(x,y,z\in A\) satisfying \((z-x)\cdot (y-x)=0\) is at most \(\left( {\begin{array}{c}n+q\\ q-1\end{array}}\right) +3\). For sufficiently large n, our result significantly improves the previous upper bound of Bennett (Eur J Combin 70:155–163, 2018), who showed that \(|A|={\mathcal {O}}(q^{\frac{n+2}{3}})\).

Popular Progression Differences in Vector Spaces

Abstract Green proved an arithmetic analog of Szemerédi’s celebrated regularity lemma and used it to verify a conjecture of Bergelson, Host, and Kra, which sharpens Roth’s theorem on three-term arithmetic progressions in dense sets. It shows that for every subset of $\mathbb{F}_p^n$ with $n$ sufficiently large, the density of three-term arithmetic progressions with some nonzero common difference is at least the random bound (the cube of the set density) up to an additive $\epsilon $. For a fixed odd prime $p$, we prove that the required dimension grows as an exponential tower of $p$’s of height $\Theta (\log (1/\epsilon ))$. This improves both the lower and upper bound, and is the first example of a result where a tower-type bound coming from applying a regularity lemma is shown to be necessary.

Erdős-Ginzburg-Ziv Constants by Avoiding Three-Term Arithmetic Progressions

For a finite abelian group $G$, The Erdős-Ginzburg-Ziv constant $\mathfrak{s}(G)$ is the smallest $s$ such that every sequence of $s$ (not necessarily distinct) elements of $G$ has a zero-sum subsequence of length $\operatorname{exp}(G)$. For a prime $p$, let $r(\mathbb{F}_p^n)$ denote the size of the largest subset of $\mathbb{F}_p^n$ without a three-term arithmetic progression. Although similar methods have been used to study $\mathfrak{s}(G)$ and $r(\mathbb{F}_p^n)$, no direct connection between these quantities has previously been established. We give an upper bound for $\mathfrak{s}(G)$ in terms of $r(\mathbb{F}_p^n)$ for the prime divisors $p$ of $\operatorname{exp}(G)$. For the special case $G=\mathbb{F}_p^n$, we prove $\mathfrak{s}(\mathbb{F}_p^n)\leq 2p\cdot r(\mathbb{F}_p^n)$. Using the upper bounds for $r(\mathbb{F}_p^n)$ of Ellenberg and Gijswijt, this result improves the previously best known upper bounds for $\mathfrak{s}(\mathbb{F}_p^n)$ given by Naslund.

The Erdős–Ginzburg–Ziv constant and progression-free subsets

Ellenberg and Gijswijt gave recently a new exponential upper bound for the size of three-term arithmetic progression free sets in (Zp)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.

On Side Lengths of Corners in Positive Density Subsets of the Euclidean Space

We generalize a result by Cook, Magyar, and Pramanik [3] on three-term arithmetic progressions in subsets of $\mathbb{R}^d$ to corners in subsets of $\mathbb{R}^d\times\mathbb{R}^d$. More precisely, if $1<p<\infty$, $p\neq 2$, and $d$ is large enough, we show that an arbitrary measurable set $A\subseteq\mathbb{R}^d\times\mathbb{R}^d$ of positive upper Banach density contains corners $(x,y)$, $(x+s,y)$, $(x,y+s)$ such that the $\ell^p$-norm of the side $s$ attains all sufficiently large real values. Even though we closely follow the basic steps from [3], the proof diverges at the part relying on harmonic analysis. We need to apply a higher-dimensional variant of a multilinear estimate from [5], which we establish using the techniques from [5] and [6].

UPPER BOUNDS FOR SUNFLOWER-FREE SETS

A collection of $k$ sets is said to form a $k$-sunflower, or $\unicode[STIX]{x1D6E5}$-system, if the intersection of any two sets from the collection is the same, and we call a family of sets ${\mathcal{F}}$sunflower-free if it contains no $3$-sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt (‘On large subsets of $\mathbb{F}_{q}^{n}$ with no three-term arithmetic progression’, Ann. of Math. (2) 185 (2017), 339–343); (‘Progression-free sets in $\mathbb{Z}_{4}^{n}$ are exponentially small’, Ann. of Math. (2) 185 (2017), 331–337) we apply the polynomial method directly to Erdős–Szemerédi sunflower problem (Erdős and Szemerédi, ‘Combinatorial properties of systems of sets’, J. Combin. Theory Ser. A 24 (1978), 308–313) and prove that any sunflower-free family ${\mathcal{F}}$ of subsets of $\{1,2,\ldots ,n\}$ has size at most $$\begin{eqnarray}|{\mathcal{F}}|\leqslant 3n\mathop{\sum }_{k\leqslant n/3}\binom{n}{k}\leqslant \left(\frac{3}{2^{2/3}}\right)^{n(1+o(1))}.\end{eqnarray}$$ We say that a set $A\subset (\mathbb{Z}/D\mathbb{Z})^{n}=\{1,2,\ldots ,D\}^{n}$ for $D&gt;2$ is sunflower-free if for every distinct triple $x,y,z\in A$ there exists a coordinate $i$ where exactly two of $x_{i},y_{i},z_{i}$ are equal. Using a version of the polynomial method with characters $\unicode[STIX]{x1D712}:\mathbb{Z}/D\mathbb{Z}\rightarrow \mathbb{C}$ instead of polynomials, we show that any sunflower-free set $A\subset (\mathbb{Z}/D\mathbb{Z})^{n}$ has size $$\begin{eqnarray}|A|\leqslant c_{D}^{n}\end{eqnarray}$$ where $c_{D}=\frac{3}{2^{2/3}}(D-1)^{2/3}$. This can be seen as making further progress on a possible approach to proving the Erdős and Rado sunflower conjecture (‘Intersection theorems for systems of sets’,J. Lond. Math. Soc. (2) 35 (1960), 85–90), which by the work of Alon et al. (‘On sunflowers and matrix multiplication’, Comput. Complexity22 (2013), 219–243; Theorem 2.6) is equivalent to proving that $c_{D}\leqslant C$ for some constant $C$ independent of $D$.

On large subsets of $\mathbb{F}_q^n$ with no three-term arithmetic progression

In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of F n q Fqn with no three terms in arithmetic progression by c n cn with c<q c<q . For q=3 q=3 , the problem of finding the largest subset of F n 3 F3n with no three terms in arithmetic progression is called the cap set problem. Previously the best known upper bound for the affine cap problem, due to Bateman and Katz, was on order n −1−ϵ 3 n n−1−ϵ3n .

Nonlinear large deviations

We present a general technique for computing large deviations of nonlinear functions of independent Bernoulli random variables. The method is applied to compute the large deviation rate functions for subgraph counts in sparse random graphs. Previous technology, based on Szemerédi's regularity lemma, works only for dense graphs. Applications are also made to exponential random graphs and three-term arithmetic progressions in random sets of integers.

On Systems of Complexity One in the Primes

AbstractConsider a translation-invariant system of linear equationsVx= 0 of complexity one, whereVis an integerr×tmatrix. We show that ifAis a subset of the primes up toNof density at leastC(log logN)–1/25t, there exists a solutionx∈ AttoVx= 0 with distinct coordinates. This extends a quantitative result of Helfgott and de Roton for three-term arithmetic progressions, while the qualitative result is known to hold for all translation-invariant systems of finite complexity by the work of Green and Tao.

A quantitative improvement for Roth's theorem on arithmetic progressions: Table 1.

We improve the quantitative estimate for Roth's theorem on three-term arithmetic progressions, showing that if $A\subset\{1,\ldots,N\}$ contains no non-trivial three-term arithmetic progressions then $\lvert A\rvert\ll N(\log\log N)^4/\log N$. By the same method we also improve the bounds in the analogous problem over $\mathbb{F}_q[t]$ and for the problem of finding long arithmetic progressions in a sumset.

ARITHMETIC PROGRESSIONS IN SETS OF SMALL DOUBLING

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.