3-term Arithmetic Progressions Research Articles

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.

Perfect powers that are sums of squares in a three term arithmetic progression

We determine primitive solutions to the equation [Formula: see text] for [Formula: see text], making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.

Gaussian Width Bounds with Applications to Arithmetic Progressions in Random Settings

Motivated by problems on random differences in Szemer\'{e}di's theorem and on large deviations for arithmetic progressions in random sets, we prove upper bounds on the Gaussian width of point sets that are formed by the image of the $n$-dimensional Boolean hypercube under a mapping $\psi:\mathbb{R}^n\to\mathbb{R}^k$, where each coordinate is a constant-degree multilinear polynomial with 0-1 coefficients. We show the following applications of our bounds. Let $[\mathbb{Z}/N\mathbb{Z}]_p$ be the random subset of $\mathbb{Z}/N\mathbb{Z}$ containing each element independently with probability $p$. $\bullet$ A set $D\subseteq \mathbb{Z}/N\mathbb{Z}$ is $\ell$-intersective if any dense subset of $\mathbb{Z}/N\mathbb{Z}$ contains a proper $(\ell+1)$-term arithmetic progression with common difference in $D$. Our main result implies that $[\mathbb{Z}/N\mathbb{Z}]_p$ is $\ell$-intersective with probability $1 - o(1)$ provided $p \geq \omega(N^{-\beta_\ell}\log N)$ for $\beta_\ell = (\lceil(\ell+1)/2\rceil)^{-1}$. This gives a polynomial improvement for all $\ell \ge 3$ of a previous bound due to Frantzikinakis, Lesigne and Wierdl, and reproves more directly the same improvement shown recently by the authors and Dvir. $\bullet$ Let $X_k$ be the number of $k$-term arithmetic progressions in $[\mathbb{Z}/N\mathbb{Z}]_p$ and consider the large deviation rate $\rho_k(\delta) = \log\Pr[X_k \geq (1+\delta)\mathbb{E}X_k]$. We give quadratic improvements of the best-known range of $p$ for which a highly precise estimate of $\rho_k(\delta)$ due to Bhattacharya, Ganguly, Shao and Zhao is valid for all odd $k \geq 5$. We also discuss connections with error correcting codes (locally decodable codes) and the Banach-space notion of type for injective tensor products of $\ell_p$-spaces.

Characters of independent Stanley sequences

Odlyzko and Stanley introduced a greedy algorithm for constructing infinite sequences with no 3-term arithmetic progressions when beginning with a finite set with no 3-term arithmetic progressions. The sequences constructed from this procedure are known as Stanley sequences and appear to have two distinct growth rates which dictate whether the sequences are structured or chaotic. A large subclass of sequences of the former type is independent sequences, which have a self-similar structure. An attribute of interest for independent sequences is the character. In this paper, building on recent progress, we prove that every nonnegative integer λ∉{1,3,5,9,11,15} is attainable as the character of an independent Stanley sequence, thus resolving a conjecture of Rolnick.

Stanley sequences with odd character

Given a set of integers containing no 3-term arithmetic progressions, one constructs a Stanley sequence by choosing integers greedily without forming such a progression. Independent Stanley sequences are a well-structured class of Stanley sequences with two main parameters: the character $\lambda(A)$ and the repeat factor $\rho(A)$. Rolnick conjectured that for every $\lambda \in \mathbb{N}_0\backslash\{1, 3, 5, 9, 11, 15\}$, there exists an independent Stanley sequence $S(A)$ such that $\lambda(A) =\lambda$. This paper demonstrates that $\lambda(A) \not\in \{1, 3, 5, 9, 11, 15\}$ for any independent Stanley sequence $S(A)$.

THE GOLDBACH PROBLEM FOR PRIMES THAT ARE SUMS OF TWO SQUARES PLUS ONE

We study the Goldbach problem for primes represented by the polynomial $x^2+y^2+1$. The set of such primes is sparse in the set of all primes, but the infinitude of such primes was established by Linnik. We prove that almost all even integers $n$ satisfying certain necessary local conditions are representable as the sum of two primes of the form $x^2+y^2+1$. This improves a result of Matom\"aki, which tells that almost all even $n$ satisfying a local condition are the sum of one prime of the form $x^2+y^2+1$ and one generic prime. We also solve the analogous ternary Goldbach problem, stating that every large odd $n$ is the sum of three primes represented by our polynomial. As a byproduct of the proof, we show that the primes of the form $x^2+y^2+1$ contain infinitely many three term arithmetic progressions, and that the numbers $\alpha p \pmod 1$ with $\alpha$ irrational and $p$ running through primes of the form $x^2+y^2+1$, are distributed rather uniformly.

Permutations that Destroy Arithmetic Progressions in Elementary $p$-Groups

Given an abelian group $G$, it is natural to ask whether there exists a permutation $\pi$ of $G$ that "destroys" all nontrivial 3-term arithmetic progressions (APs), in the sense that $\pi(b) - \pi(a) \neq \pi(c) - \pi(b)$ for every ordered triple $(a,b,c) \in G^3$ satisfying $b-a = c-b \neq 0$. This question was resolved for infinite groups $G$ by Hegarty, who showed that there exists an AP-destroying permutation of $G$ if and only if $G/\Omega_2(G)$ has the same cardinality as $G$, where $\Omega_2(G)$ denotes the subgroup of all elements in $G$ whose order divides $2$. In the case when $G$ is finite, however, only partial results have been obtained thus far. Hegarty has conjectured that an AP-destroying permutation of $G$ exists if $G = \mathbb{Z}/n\mathbb{Z}$ for all $n \neq 2,3,5,7$, and together with Martinsson, he has proven the conjecture for all $n > 1.4 \times 10^{14}$. In this paper, we show that if $p$ is a prime and $k$ is a positive integer, then there is an AP-destroying permutation of the elementary $p$-group $(\mathbb{Z}/p\mathbb{Z})^k$ if and only if $p$ is odd and $(p,k) \not\in \{(3,1),(5,1), (7,1)\}$.

Covering the Large Spectrum and Generalized Riesz Products

Chang's lemma is a widely employed result in additive combinatorics. It gives bounds on the dimension of the large spectrum of probability distributions on finite abelian groups. Recently, Bloom (2016) presented a powerful variant of Chang's lemma that yields the strongest known quantitative version of Roth's theorem on 3-term arithmetic progressions in dense subsets of the integers. In this note, we show how such theorems can be derived from the approximation of probability measures via entropy maximization.

On the Number of Similar Instances of a Pattern in a Finite Set

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is shown to be no more than $\lfloor{(4 n-1)(n-1)/18}\rfloor$. The number of $k$-term arithmetic progressions that lie within an $n$-point subset of the line is shown to be at most $(n-r)(n+r-k+1)/(2 k-2)$, where $r$ is the remainder when $n$ is divided by $k-1$. This upper bound is achieved when the $n$ points themselves form an arithmetic progression, but for some values of $k$ and $n$, it can also be achieved for other configurations of the $n$ points, and a full classification of such optimal configurations is given. These results are achieved using a new general method based on ordering relations.

The Number of Subsets of Integers with Nok-Term Arithmetic Progression

Addressing a question of Cameron and Erd\Ho s, we show that, for infinitely many values of $n$, the number of subsets of $\{1,2,\ldots, n\}$ that do not contain a $k$-term arithmetic progression is at most $2^{O(r_k(n))}$, where $r_k(n)$ is the maximum cardinality of a subset of $\{1,2,\ldots, n\}$ without a $k$-term arithmetic progression. This bound is optimal up to a constant factor in the exponent. For all values of $n$, we prove a weaker bound, which is nevertheless sufficient to transfer the current best upper bound on $r_k(n)$ to the sparse random setting. To achieve these bounds, we establish a new supersaturation result, which roughly states that sets of size $\Theta(r_k(n))$ contain superlinearly many $k$-term arithmetic progressions. For integers $r$ and $k$, Erd\Ho s asked whether there is a set of integers $S$ with no $(k+1)$-term arithmetic progression, but such that any $r$-coloring of $S$ yields a monochromatic $k$-term arithmetic progression. Nesetřil and Rodl, and independently Spencer, answered this question affirmatively. We show the following density version: for every $k\ge 3$ and $\delta>0$, there exists a reasonably dense subset of primes $S$ with no $(k+1)$-term arithmetic progression, yet every $U\subseteq S$ of size $|U|\ge\delta|S|$ contains a $k$-term arithmetic progression. Our proof uses the hypergraph container method, which has proven to be a very powerful tool in extremal combinatorics. The idea behind the container method is to have a small certificate set to describe a large independent set. We give two further applications in the appendix using this idea.

Well Ordering Groups with no Monotone Arithmetic Progressions

Karolyi–Kos and Ardal–Brown–Jungic proved that every vector space over \(\mathbb {Q}\) has an ordering with no monotone three term arithmetic progression (3-AP). We show that every solvable group has a well ordering with no monotone 6-AP, and each hypoabelian group has an ordering omitting monotone 5-APs. Finally, we prove that every group has a well ordering with no infinite monotone AP.

Narrow Arithmetic Progressions in the Primes

We study arithmetic progressions in primes with common differences as small as possible. Tao and Ziegler showed that, for any $k \geq 3$ and $N$ large, there exist non-trivial $k$-term arithmetic progressions in (any positive density subset of) the primes up to $N$ with common difference $O((\log N)^{L_k})$, for an unspecified constant $L_k$. In this work we obtain this statement with the precise value $L_k = (k-1) 2^{k-2}$. This is achieved by proving a relative version of Szemer\'{e}di's theorem for narrow progressions requiring simpler pseudorandomness hypotheses in the spirit of recent work of Conlon, Fox, and Zhao.

Finite configurations in sparse sets

Let E ⊆ ℝ n be a closed set of Hausdorff dimension α. For m > n, let{B 1, …, B k } be n × (m − n) matrices. We prove that if the system of matrices B j is non-degenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of m depending on n and k, the set E contains a translate of a non-trivial k-point configuration {B 1 y, …, B k y}. As a consequence, we are able to establish existence of certain geometric configurations in Salem sets (such as parallelograms in ℝ n and isosceles right triangles in ℝ2). This can be viewed as a multidimensional analogue of the result of [25] on 3-term arithmetic progressions in subsets of ℝ.

Permutations Destroying Arithmetic Progressions in Finite Cyclic Groups

A permutation $\pi$ of an abelian group $G$ is said to destroy arithmetic progressions (APs) if, whenever $(a, \, b, \, c)$ is a non-trivial 3-term AP in $G$, that is $c-b=b-a$ and $a, \, b, \, c$ are not all equal, then $(\pi(a), \, \pi(b), \pi(c))$ is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of $\mathbb{Z}_n$, for all $n \not\in \{2, \, 3, \, 5, \, 7\}$. Here we prove, as a special case of a more general result, that such a permutation exists for all $n \geq n_0$, for some explicitly constructed number $n_0 \approx 1.4 \times 10^{14}$. We also construct such a permutation of $\mathbb{Z}_p$ for all primes $p > 3$ such that $p \equiv 3 \; ({\hbox{mod $8$}})$.

Novel structures in Stanley sequences

Given a set of integers with no 3-term arithmetic progression, one constructs a Stanley sequence by choosing integers greedily without forming such a progression. This paper offers two main contributions to the theory of Stanley sequences. First, we describe all known Stanley sequences with closed-form expressions as solutions to constraints in modular arithmetic, defining the modular and pseudomodular Stanley sequences. Second, we introduce the basic Stanley sequences, whose elements arise as the sums of finite subsets of a basis sequence. Applications of our results include the construction of Stanley sequences with arbitrarily large gaps between terms, answering a weak version of a problem by Erdős et al. Finally, we generalize several results about Stanley sequences to p-free sequences, where p is any odd prime.

Arithmetic progressions with a pseudorandom step

Let α,σ>0 and let A and S be subsets of a finite abelian group G of densities α and σ, respectively, both independent of |G|. Without any additional restrictions, the set A need not contain a 3-term arithmetic progression whose common gap lies in S. What is then the weakest pseudorandomness assumption that if put on S would imply that A contains such a pattern?More precisely, what is the least integer k≥2 for which there exists an η=η(α,σ) such that ‖S−σ‖Uk(G)≤η implies that A contains a non-trivial 3-term arithmetic progression with a common gap in S? Here, ‖⋅‖Uk(G) denotes the kth Gowers norm.For G=Zn we observe that k must be at least 3. However for G=Fpn we show that k=2 is sufficient, where here p is an odd prime and n is sufficiently large.

On sets of integers which contain no three terms in geometric progression

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. This construction is optimal in the sense that our method gives a way of effectively computing the greatest possible upper density of a geometric-progression-free set. We also show that geometric progressions in Z/nZ behave more like Roth's theorem in that one cannot take any fixed positive proportion of the integers modulo a sufficiently large value of n while avoiding geometric progressions.

A relative Szemerédi theorem

The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemer\'edi theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmetic progressions. In this paper, we give a simple proof of a strengthening of the relative Szemer\'edi theorem, showing that a much weaker pseudorandomness condition is sufficient. Our strengthened version can be applied to give the first relative Szemer\'edi theorem for $k$-term arithmetic progressions in pseudorandom subsets of $\mathbb{Z}_N$ of density $N^{-c_k}$. The key component in our proof is an extension of the regularity method to sparse pseudorandom hypergraphs, which we believe to be interesting in its own right. From this we derive a relative extension of the hypergraph removal lemma. This is a strengthening of an earlier theorem used by Tao in his proof that the Gaussian primes contain arbitrarily shaped constellations and, by standard arguments, allows us to deduce the relative Szemer\'edi theorem.

Counting independent sets in graphs

In this short survey article, we present an elementary, yet quite powerful, method of enumerating independent sets in graphs. This method was first employed more than three decades ago by Kleitman and Winston and has subsequently been used numerous times by many researchers in various contexts. Our presentation of the method is illustrated with several applications of it to ‘real-life’ combinatorial problems. In particular, we derive bounds on the number of independent sets in regular graphs, sum-free subsets of {1,…,n}, and C4-free graphs and give a short proof of an analogue of Roth’s theorem on 3-term arithmetic progressions in sparse random sets of integers which was originally formulated and proved by Kohayakawa, Łuczak, and Rödl.