Sequences of Integers Avoiding 3-term Arithmetic Progressions

We study the equation (x−4r)3+(x−3r)3+(x−2r)3+(x−r)3+x3+(x+r)3+(x+2r)3+(x+3r)3+(x+4r)3=yp, which is a natural continuation of previous works carried out by A. Argáez-García and the fourth author (perfect powers that are sums of cubes of a three, five and seven term arithmetic progression). Under the assumptions 0<r≤106, p≥5 a prime and gcd(x,r)=1, we show that solutions must satisfy xy=0. Moreover, we study the equation for prime exponents 2 and 3 in greater detail. Under the assumptions r>0 a positive integer and gcd(x,r)=1 we show that there are infinitely many solutions for p=2 and p=3 via explicit constructions using integral points on elliptic curves. We use an amalgamation of methods in computational and algebraic number theory to overcome the increased computational challenge. Most notable is a significant computational efficiency obtained through appealing to Bilu, Hanrot and Voutier’s Primitive Divisor Theorem and the method of Chabauty, as well as employing a Thue equation solver earlier on.

Two well-studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression or in geometric progression. We study generalizations of this problem by varying the kinds of progressions to be avoided and the metrics used to evaluate the density of the resulting subsets. One can view a 3-term arithmetic progression as a sequence \(x, f_n(x), f_n(f_n(x))\), where \(f_n(x) = x + n\), n a nonzero integer. Thus, avoiding 3-term arithmetic progressions are equivalent to containing no three elements of the form \(x, f_n(x), f_n(f_n(x))\) with \(f_n \in \mathscr {F}_\mathrm{t}\), the set of integer translations. One can similarly construct related progressions using different families of functions. We investigate several such families, including geometric progressions (\(f_n(x) = nx\) with \(n > 1\) a natural number) and exponential progressions (\(f_n(x) = x^n\)). Progression-free sets are often constructed “greedily,” including every number so long as it is not in progression with any of the previous elements. Rankin characterized the greedy geometric-progression-free set in terms of the greedy arithmetic set. We characterize the greedy exponential set and prove that it has asymptotic density 1 and then discuss how the optimality of the greedy set depends on the family of functions used to define progressions. Traditionally, the size of a progression-free set is measured using the (upper) asymptotic density; however, we consider several different notions of density, including the uniform and exponential densities.

It is shown that there is a subsetS of integers containing no (k+1)-term arithmetic progression such that if the elements ofS are arbitrarily colored (any number of colors),S will contain ak-term arithmetic progression for which all of its terms have the same color, or all have distinct colors.

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.

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.

Fix integers $r \ge 2$ and $1\le s_1\le \cdots \le s_{r-1}\le t$ and set $s=\prod _{i=1}^{r-1}s_i$ . Let $K=K(s_1, \ldots , s_{r-1}, t)$ denote the complete $r$ -partite $r$ -uniform hypergraph with parts of size $s_1, \ldots , s_{r-1}, t$ . We prove that the Zarankiewicz number $z(n, K)= n^{r-1/s-o(1)}$ provided $t\gt 3^{s+o(s)}$ . Previously this was known only for $t \gt ((r-1)(s-1))!$ due to Pohoata and Zakharov. Our novel approach, which uses Behrend’s construction of sets with no 3-term arithmetic progression, also applies for small values of $s_i$ , for example, it gives $z(n, K(2,2,7))=n^{11/4-o(1)}$ where the exponent 11/4 is optimal, whereas previously this was only known with 7 replaced by 721.

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 $S(A)$.

Let ${\cal P}_f({\bf N})$ be the set of finite nonempty subsets of ${\bf N}$ and for $F,G\in{\cal P}_f({\bf N})$ write $F &lt; G$ when $\max F &lt; \min G$. Let $X=\{(F,G):F,G\in{\cal P}_f({\bf N})$ and $F &lt; G\}$. A triangle in $X$ is a set of the form $\{(F\cup H,G),(F,G),(F,H\cup G)\}$ where $F &lt; H &lt; G$. Motivated by a question of Erdős, Nešetríl, and Rödl regarding three term arithmetic progressions, we show that any finite subset $Y$ of $X$ contains a relatively large triangle free subset. Exact values are obtained for the largest triangle free sets which can be guaranteed to exist in any set $Y\subseteq X$ with $n$ elements for all $n\leq 14$.

The pattern avoidance problem seeks to construct a set with large fractal dimension that avoids a prescribed pattern, such as three term arithmetic progressions, or more general patterns, such as finding a set whose Cartesian product avoids the zero set of a given function. Previous work on the subject has considered patterns described by polynomials, or functions satisfying certain regularity conditions. We provide an exposition of some results in this setting, as well as considering new strategies to avoid what we call `rough patterns'. This thesis contains an expanded description of a method described in a previous paper by the author and his collaborators Malabika Pramanik and Joshua Zahl, as well as new results in the rough pattern avoidance setting. There are several problems that fit into the pattern of rough pattern avoidance. For instance, we prove that for any set $X$ with lower Minkowski dimension $s$, there exists a set $Y$ with Hausdorff dimension $1-s$ such that for any rational numbers $a_1, \dots, a_N$, the set $a_1 Y + \dots + a_N Y$ is disjoint from $X$, or intersects with $X$ solely at the origin. As a second application, we construct subsets of Lipschitz curves with dimension $1/2$ not containing the vertices of any isosceles triangle.

Consider a set $A$ with no $p$-term arithmetic progressions for $p$ prime. The $p$-Stanley sequence of a set $A$ is generated by greedily adding successive integers that do not create a $p$-term arithmetic progression. For $p>3$ prime, we give two distinct constructions for $p$-Stanley sequences which have a regular structure and satisfy certain conditions in order to be modular $p$-Stanley sequences, a set of particularly nice sequences defined by Moy and Rolnick which always have a regular structure. Odlyzko and Stanley conjectured that the 3-Stanley sequence generated by $\{0,n\}$ only has a regular structure if $n=3^k$ or $n=2\cdot 3^k$. For $p>3$ we find a substantially larger class of integers $n$ such that the $p$-Stanley sequence generated from $\{0,n\}$ is a modular $p$-Stanley sequence and numerical evidence given by Moy and Rolnick suggests that these are the only $n$ for which the $p$-Stanley sequence generated by $\{0,n\}$ is a modular $p$-Stanley sequence. Our second class is a generalization of a construction of Rolnick for $p=3$ and is thematically similar to the analogous construction by Rolnick.

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.

This thesis presents some contributions in additive combinatorics and arithmetic Ramsey theory. More specifically, it deals with the interaction between combinatorics, number theory and additive combinatorics. This area saw a great improvement with the Szemerédi Regularity Lemma and some of the results that followed. The Regularity Lemma and its consequences have become a widely used tool in graph theory, combinatorics and number theory. Furthermore, its language and point of view has deeply changed the face of additive number theory, a fact universally acknowledged by the Abel award given to Szemerédi in 2012. One of the main reasons for the prize has been Szemerédi's theorem, a result regarding the existence of arbitrarily long arithmetic progressions in dense sets of the integers, the proof of which uses the Regularity Lemma in a key step. One of the earlier consequences of the Regularity Lemma was the Removal Lemma for graphs that was used by Ruzsa and Szemerédi to show Roth theorem, regarding the existence of 3-term arithmetic progressions in dense sets of the integers, in a combinatorial way. The Removal Lemma states that in any graph K with few copies of a subgraph, say a triangle, we can remove few edges from K so that the result contains no copy of the subgraph. This has become a key tool in the applications of the so-called Regularity Method, which has extensive literature in combinatorics, graph theory, number theory and computer science. In 2005 Green introduced a regularity lemma for Abelian groups as well as an algebraic removal lemma. The removal lemma for groups states that, for a given finite Abelian group G, if there are o(|G|^3) solution to x+y+z+t=0 with the variables taking values in S, a subset of G, then we can remove o(|G|) elements from S to make the set S solution-free. The main contributions of this work corresponds to extensions of the removal lemma for groups to either more general contexts, like non-necessary Abelian finite groups, or to linear systems of equations for finite Abelian groups. The main goal is to give a comprehensive and more general framework for many results in additive number theory like Szemerédi Theorem. In particular, we show that the removal lemma for groups by Green can be extended to non-necessary Abelian finite groups. Moreover, we prove a removal lemma for linear systems on finite fields: for every e&gt;0 there exists a d&gt;0 such that if A is a (k x m) linear system of equations with coefficients in a finite field F and the number of solutions to Ax=b, where each variable takes values from a subset Si in F is less than d times |F| raised to m-k, then by removing less than e|F| elements in each Si we can make the resulting sets solution-free, thus solving a conjecture by Green to that respect. Even more, if A is an integer linear system, G is a finite Abelian group, and the determinantal of A and |G| are coprime, then a similar statement holds. Let us mention that the last result allows us to characterize those linear systems where any set S with size proportional to G has a nontrivial solution in S, provided |G| is large enough. This extends the validity of Szmerédi's theorem to finite Abelian groups. These extensions of the removal lemma have been used in arithmetic Ramsey theory to obtain counting results for the number of monochromatic solutions of linear systems. The main result from a work by Frankl, Graham and Rödl in '88 states that the number of monochromatic solutions of regular systems in integer intervals is in fact a positive proportion of the total number of solutions. We give analogous results for solutions in Abelian groups with bounded exponent, for which the main tool in the torsion-free case cannot be applied. Density versions of these counting results are also obtained, in this case with a full characterization.

In this note we prove that there is a linear ordering of the set of real numbers for which there is no monotonic 3-term arithmetic progression. This answers the question (asked by Erdős and Graham) of whether or not every linear ordering of the reals must have a monotonic k-term arithmetic progression for every k.

The set {1, 25, 49} is a 3-term collection of integers which forms an arithmetic progression of perfect squares. We view the set {(1, 1), (5, 25), (7, 49)} as a 3-term collection of rational points on the parabola y = x2whose y-coordinates form an arithmetic progression. In this exposition, we provide a generalization to 3-term arithmetic progressions on arbitrary conic sections [Formula: see text] with respect to a linear rational map [Formula: see text]. We explain how this construction is related to rational points on the universal elliptic curve Y2+ 4XY + 4kY = X3+ kX2classifying those curves possessing a rational 4-torsion point.