3-term Arithmetic Progressions Research Articles

From discrete to continuous: Monochromatic 3-term arithmetic progressions

We prove a known 2-coloring of the integers [ N ] ≔ { 1 , 2 , 3 , … , N } [N] ≔\{1,2,3,\dots ,N\} minimizes the number of monochromatic arithmetic 3-progressions under certain restrictions. A monochromatic arithmetic progression is a set of equally-spaced integers that are all the same color. Previous work by Parrilo, Robertson and Saracino conjectured an optimal coloring for large N N that involves 12 12 colored blocks. Here, we prove that the conjectured coloring minimizes monochromatic arithmetic 3 3 -progressions among anti-symmetric colorings with 12 12 or fewer colored blocks. We leverage a connection to the coloring of the continuous interval [ 0 , 1 ] [0,1] used by Parrilo, Robertson, and Saracino as well as by Butler, Costello and Graham. Our proof identifies classes of colorings with permutations, then counts the permutations using mixed integer linear programming.

On the Size of Subsets of $\mathbb{F}_q^n$ Avoiding Solutions to Linear Systems with Repeated Columns

Consider a system of $m$ balanced linear equations in $k$ variables with coefficients in $\mathbb{F}_q$. If $k \geq 2m + 1$, then a routine application of the slice rank method shows that there are constants $\beta,\gamma \geq 1$ with $\gamma < q$ such that, for every subset $S \subseteq \mathbb{F}_q^n$ of size at least $\beta \cdot \gamma^n$, the system has a solution $(x_1,\ldots,x_k) \in S^k$ with $x_1,\ldots,x_k$ not all equal. Building on a series of papers by Mimura and Tokushige and on a paper by Sauermann, this paper investigates the problem of finding a solution of higher non-degeneracy; that is, a solution where $x_1,\ldots,x_k$ are pairwise distinct, or even a solution where $x_1,\ldots,x_k$ do not satisfy any balanced linear equation that is not a linear combination of the equations in the system.
 In this paper, we focus on linear systems with repeated columns. For a large class of systems of this type, we prove that there are constants $\beta,\gamma \geq 1$ with $\gamma < q$ such that every subset $S \subseteq \mathbb{F}_q^n$ of size at least $\beta \cdot \gamma^n$ contains a solution that is non-degenerate (in one of the two senses described above). This class is disjoint from the class covered by Sauermann's result, and captures the systems studied by Mimura and Tokushige into a single proof. Moreover, a special case of our results shows that, if $S \subseteq \mathbb{F}_p^n$ is a subset such that $S - S$ does not contain a non-trivial $k$-term arithmetic progression (with $p$ prime and $3 \leq k \leq p$), then $S$ must have exponentially small density.

Anti-van der Waerden Numbers on Graphs

In this paper arithmetic progressions on the integers and the integers modulo n are extended to graphs. A k-term arithmetic progression of a graph G (k-AP) is a list of k distinct vertices such that the distance between consecutive pairs is constant. A rainbow k-AP is a k-AP where each vertex is colored distinctly. This allows for the definition of the anti-van der Waerden number of a graph G, which is the least positive integer r such that every exact r-coloring of G contains a rainbow k-AP. Much of the focus of this paper is on 3-term arithmetic progressions for which general bounds are obtained based on the radius and diameter of a graph. The general bounds are improved for trees and Cartesian products and exact values are determined for some classes of graphs. Longer k-term arithmetic progressions are considered and a connection between the Ramsey number of paths and the anti-van der Waerden number of graphs is established.Please confirm if the inserted city and country name for all affiliations is correct. Amend if necessary.The cities and affiliations are correct.

Integer Colorings with No Rainbow 3-Term Arithmetic Progression

In this paper, we study the rainbow Erdős-Rothschild problem with respect to 3-term arithmetic progressions. We obtain the asymptotic number of $r$-colorings of $[n]$ without rainbow 3-term arithmetic progressions, and we show that the typical colorings with this property are 2-colorings. We also prove that $[n]$ attains the maximum number of rainbow 3-term arithmetic progression-free $r$-colorings among all subsets of $[n]$. Moreover, the exact number of rainbow 3-term arithmetic progression-free $r$-colorings of $\mathbb{Z}_p$ is obtained, where $p$ is any prime and $\mathbb{Z}_p$ is the cyclic group of order $p$.

A Szemerédi-type theorem for subsets of the unit cube

We investigate gaps of $n$-term arithmetic progressions $x, x+y, \ldots, x+(n-1)y$ inside a positive measure subset $A$ of the unit cube $[0,1]^d$. If lengths of their gaps $y$ are evaluated in the $\ell^p$-norm for any $p$ other than $1, 2, \ldots, n-1$, and $\infty$, and if the dimension $d$ is large enough, then we show that the numbers $\|y\|_{\ell^p}$ attain all values from an interval, the length of which depends only on $n$, $p$, $d$, and the measure of $A$. Known counterexamples prevent generalizations of this result to the remaining values of the exponent $p$. We also give an explicit bound for the length of the aforementioned interval. The proof makes the bound depend on the currently available bounds in Szemer\'{e}di's theorem on the integers, which are used as a black box. A key ingredient of the proof are power-type cancellation estimates for operators resembling the multilinear Hilbert transforms. As a byproduct of the approach we obtain a quantitative improvement of the corresponding (previously known) result for side lengths of $n$-dimensional cubes with vertices lying in a positive measure subset of $([0,1]^2)^n$.

Local limit theorems for subgraph counts

We introduce a general framework for studying anticoncentration and local limit theorems for random variables, including graph statistics. Our methods involve an interplay between Fourier analysis, decoupling, hypercontractivity of Boolean functions, and transference between ``fixed-size'' and ``independent'' models. We also adapt a notion of ``graph factors'' due to Janson. As a consequence, we derive a local central limit theorem for connected subgraph counts in the Erd\H{o}s-Renyi random graph $G(n,p)$, building on work of Gilmer and Kopparty and of Berkowitz. These results improve an anticoncentration result of Fox, Kwan, and Sauermann and partially answers a question of Fox, Kwan, and Sauermann. We also derive a local limit central limit theorem for induced subgraph counts, as long as $p$ is bounded away from a set of ``problematic'' densities, partially answering a question of Fox, Kwan, and Sauermann. We then prove these restrictions are necessary by exhibiting a disconnected graph for which anticoncentration for subgraph counts at the optimal scale fails for all constant $p$, and finding a graph $H$ for which anticoncentration for induced subgraph counts fails in $G(n,1/2)$. These counterexamples resolve anticoncentration conjectures of Fox, Kwan, and Sauermann in the negative. Finally, we also examine the behavior of counts of $k$-term arithmetic progressions in subsets of $\mathbb{Z}/n\mathbb{Z}$ and deduce a local limit theorem wherein the behavior is Gaussian at a global scale but has nontrivial local oscillations (according to a Ramanujan theta function). These results improve on results of and answer questions of the authors and Berkowitz, and answer a question of Fox, Kwan, and Sauermann.

The Maximal Number of 3-Term Arithmetic Progressions in Finite Sets in Different Geometries

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 $$d(a,b)=d(b,c)=d(a,c)/2$$ . 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 hold in spherical geometry or in the r-regular tree, for any $$r\ge 3$$ .

Random Van der Waerden Theorem

In this paper, we prove a sparse random analogue of the Van der Waerden Theorem. We show that, for all $r > 2$ and all $q_1 \geq q_2 \geq \dotsb \geq q_r \geq 3 \in \mathbb{N}$, $n^{-\frac{q_2}{q_1(q_2-1)}}$ is a threshold for the following property: For every $r$-coloring of the $p$-random subset of $\{1,\dotsc,n\}$, there exists a monochromatic $q_i$-term arithmetic progression colored $i$, for some $i$. This extends the results of Rödl and Ruciński for the symmetric case $q_1 = q_2 = \dotsb = q_r$. The proof of the 1-statement is based on the Hypergraph Container Method by Balogh, Morris and Samotij and Saxton and Thomason. The proof of the 0-statement is an extension of Rödl and Ruciński's argument for the symmetric case.

New lower bounds for van der Waerden numbers

AbstractWe show that there is a red-blue colouring of$[N]$with no blue 3-term arithmetic progression and no red arithmetic progression of length$e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number$w(3,k)$is bounded below by$k^{b(k)}$, where$b(k) = c \big ( \frac {\log k}{\log \log k} \big )^{1/3}$. Previously it had been speculated, supported by data, that$w(3,k) = O(k^2)$.

Four‐term progression free sets with three‐term progressions in all large subsets

AbstractThis 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 which does not contain a four‐term arithmetic progression, with the property that for every subset with , contains a nontrivial three term arithmetic progression. We derive this from a more general quantitative Roth‐type theorem in random subsets of , which improves a result of Kohayakawa–Łuczak–Rödl/Tao–Vu. We also discuss a similar phenomenon over the integers. Finally, we include another application of our methods, showing that for sets in or the property of “having nontrivial three‐term progressions in all large subsets” is almost entirely uncorrelated with the property of “having large additive energy.”

Threshold functions for substructures in random subsets of finite vector spaces

The study of substructures in random objects has a long history, beginning with Erdős and Renyi's work on subgraphs of random graphs. We study the existence of certain substructures in random subsets of vector spaces over finite fields. First we provide a general framework which can be applied to establish coarse threshold results and prove a limiting Poisson distribution at the threshold scale. To illustrate our framework we apply our results to $k$-term arithmetic progressions, sums, right triangles, parallelograms and affine planes. We also find coarse thresholds for the property that a random subset of a finite vector space is sum-free, or is a Sidon set.

Triple correlation sums of coefficients of cusp forms

We produce nontrivial asymptotic estimates for shifted sums of the form ∑a(h)b(m)c(2m−h), in which a(n),b(n),c(n) are un-normalized Fourier coefficients of holomorphic cusp forms. These results are unconditional, but we demonstrate how to strengthen them under the Riemann Hypothesis. As an application, we show that there are infinitely many three term arithmetic progressions n−h,n,n+h such that a(n−h)a(n)a(n+h)≠0.

Colorings with only rainbow arithmetic progressions

If we want to color $$1,2,\ldots ,n$$ with the property that all 3-term arithmetic progressions are rainbow (that is, their elements receive 3 distinct colors), then, obviously, we need to use at least n/2 colors. Surprisingly, much fewer colors suffice if we are allowed to leave a negligible proportion of integers uncolored. Specifically, we prove that there exist $$\alpha ,\beta <1$$ such that for every n, there is a subset A of $$\{1,2,\ldots ,n\}$$ of size at least $$n-n^{\alpha }$$ , the elements of which can be colored with $$n^{\beta }$$ colors with the property that every 3-term arithmetic progression in A is rainbow. Moreover, $$\beta $$ can be chosen to be arbitrarily small. Our result can be easily extended to k-term arithmetic progressions for any $$k\ge 3$$ . As a corollary, we obtain a simple proof of the following result of Alon, Moitra, and Sudakov, which can be used to design efficient communication protocols over shared directional multi-channels. There exist $$\alpha ',\beta '<2$$ such that for every n, there is a graph with n vertices and at least $$\left( {\begin{array}{c}n 2\end{array}}\right) -n^{\alpha '}$$ edges, whose edge set can be partitioned into at most $$n^{\beta '}$$ induced matchings.

On perfect powers that are sums of cubes of a seven term arithmetic progression

We prove that the equation (x−3r)3+(x−2r)3+(x−r)3+x3+(x+r)3+(x+2r)3+(x+3r)3=yp only has solutions which satisfy xy=0 for 1≤r≤106 and p≥5 prime. This article complements the work on the equations (x−r)3+x3+(x+r)3=yp in [2] and (x−2r)3+(x−r)3+x3+(x+r)3+(x+2r)3=yp in [1]. The methodology in this paper makes use of the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier for a complete resolution of the Diophantine equation.

Zero-sum analogues of van der Waerden’s theorem on arithmetic progressions

Let $r$ and $k$ be positive integers with $r \mid k$. Denote by $w_{\mathrm{\mathfrak{z}}}(k;r)$ the minimum integer such that every coloring $\chi:[1,w_{\mathrm{\mathfrak{z}}}(k;r)] \rightarrow \{0,1,\dots,r-1\}$ admits a $k$-term arithmetic progression $a,a+d,\dots,a+(k-1)d$ with $\sum_{j=0}^{k-1} \chi(a+jd) \equiv 0 \,(\mathrm{mod }\,r)$. We investigate these numbers as well as a mixed monochromatic/zero-sum analogue. We also present an interesting reciprocity between the van der Waerden numbers and $w_{\mathrm{\mathfrak{z}}}(k;r)$.

On Szemerédi’s theorem with differences from a random set

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 Fran

Abundance of progressions in a commutative semigroup by elementary means

Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise syndetic set contains k term 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. Beiglbock provided an elementary proof of the above result and asked whether the combinatorial argument in his proof can be enhanced in a way which makes it applicable to a more abstract setting. In this work we provide a positive answer to Beiglbock question for commutative semigroup.

Proof of the Brown–Erdős–Sós conjecture in groups

AbstractThe conjecture of Brown, Erdős and Sós from 1973 states that, for any k ≥ 3, if a 3-uniform hypergraph H with n vertices does not contain a set of k +3 vertices spanning at least k edges then it has o(n2) edges. The case k = 3 of this conjecture is the celebrated (6, 3)-theorem of Ruzsa and Szemerédi which implies Roth’s theorem on 3-term arithmetic progressions in dense sets of integers. Solymosi observed that, in order to prove the conjecture, one can assume that H consists of triples (a, b, ab) of some finite quasigroup Γ. Since this problem remains open for all k ≥ 4, he further proposed to study triple systems coming from finite groups. In this case he proved that the conjecture holds also for k = 4. Here we completely resolve the Brown–Erdős–Sós conjecture for all finite groups and values of k. Moreover, we prove that the hypergraphs coming from groups contain sets of size $\Theta (\sqrt k )$ which span k edges. This is best possible and goes far beyond the conjecture.

On perfect powers that are sums of cubes of a five term arithmetic progression

We prove that the equation (x−2r)3+(x−r)3+x3+(x+r)3+(x+2r)3=yp only has solutions which satisfy xy=0 for 1≤r≤106 and p≥5 prime.

On arithmetic progressions in Lucas sequences – II

In this paper, we provide an effective and practical method to find all three term arithmetic progressions in a given Lucas sequence of the first or second kind. Our interest is the case when the sequence has a negative discriminant, since the case of positive discriminant has recently been resolved by Hajdu et al. [11]. We present a conjecture on the maximal number and length of such arithmetic progressions based on computational evidence.