Abstract: We compute the catenary degree of elements contained in numerical monoids generated by arithmetic sequences. We find that this can be done by describing each element in terms of the cardinality of its length set and of its set of factorizations. As a corollary, we find for such monoids that the catenary degree becomes fixed on large elements. This allows us to define and compute the dissonance number- the largest element with a catenary degree different from the fixed value. We determine the dissonance number in terms of the arithmetic sequence’s starting point and its number of generators.

Abstract: Define $r_4(N)$ to be the largest cardinality of a set $A \subset \{1,\dots,N\}$ which does not contain four elements in arithmetic progression. In 1998 Gowers proved that \[ r_4(N) \ll N(\log \log N)^{-c}\] for some absolute constant $c>0$. In 2005, the authors improved this to \[ r_4(N) \ll N e^{-c\sqrt{\log\log N}}.\] In this paper we further improve this to \[ r_4(N) \ll N(\log N)^{-c},\] which appears to be the limit of our methods.

Abstract: Using only elementary arguments, Cassels and Uchiyama (independently) determined all squares that are sums of three consecutive cubes. Zhongfeng Zhang extended this result and determined all perfect powers that are sums of three consecutive cubes. Recently, the equation $(x-r)^k + x^k + (x+r)^k$ has been studied for $k=4$ by Zhongfeng Zhang and for $k=2$ by Koutsianas. In this paper, we complement the work of Cassels, Koutsianas and Zhang by considering the case when $k=3$ and showing that the equation $(x-r)^3+x^3+(x+r)^3=y^n$ with $n\geq 5$ a prime and $0 < r \leq 10^6$ only has trivial solutions $(x,y,n)$ which satisfy $xy=0$.

Abstract: Quantitative bounds in the polynomial Szemeredi theorem: the homogeneous case, Discrete Analysis 2017:5, 34 pp. Szemeredi's theorem, proved in 1975, asserts that for every positive integer $k$ and every $\delta>0$ there exists $n$ such that every subset $A$ of $\{1,2,\dots,n\}$ of cardinality at least $\delta n$ contains an arithmetic progression of length $k$. This apparently simple statement turns out to be surprisingly hard to prove, though it now has many different proofs and has become a hugely influential result in additive and extremal combinatorics. In 1996, Bergelson and Leibman proved a far-reaching extension of Szemeredi's theorem, which states the following. Let $P_1,\dots,P_k$ be polynomials that take integer values at the integers and have no constant terms. Then for every $\delta>0$ there exists $n$ (which depends on $\delta$ and the polynomials $P_1,\dots,P_k$) such that for every subset $A$ of $\{1,\dots,n\}$ of cardinality at least $\delta n$ there exist positive integers $a,d$ such that $a+P_1(d),\dots,a+P_k(d)$ all belong to $A$. This is a huge common generalization of Szemeredi's theorem, which is the case $P_i(d)=(i-1)d$, and a theorem of Furstenberg and Sarkozy, which is the case $P_1(d)=0$ and $P_2(d)=d^2$, so it tells us that a dense subset of $\{1,2,\dots,n\}$ must contain two elements that differ by a perfect square. Szemeredi's proof of Szemeredi's theorem was an intricate combinatorial argument that led to a very weak bound -- indeed, so weak that it has not been explicitly calculated. So although it is in principle quantitative, in effect it is a qualitative statement. In 1977 Furstenberg gave a new proof using ergodic theory, which led to many extensions of the theorem, including that of Bergelson and Leibman, but these methods were ineffective by their very nature, so did not provide bounds. (Tao later found a quantitative version of Furstenberg's proof that did give bounds, though once again they were very weak.) The first reasonable bound was due to Gowers, who gave a bound for $n$ with a doubly exponential dependence on a power of $1/\delta$, the power depending on $k$ in such a way that the dependence on $k$ for fixed $\delta$ is quintuply exponential. For a long time, the only cases of the Bergelson-Leibman theorem for which reasonable bounds were known were those where the configurations being sought were of the form $\{a,a+P(d)\}$. In 2002, Green obtained quantitative bounds for the related problem where the forbidden configurations are those of the form $\{a,a+d_1^2+d_2^2,a+2(d_1^2+d_2^2)\}$, and there matters stood for some time. In 2013, Tao wrote [an interesting blog post](https://terrytao.wordpress.com/2013/02/28/a-fourier-free-proof-of-the-furstenberg-sarkozy-theorem/) in which he described an approach that he had discovered with Green and Ziegler a few years earlier for proving the Furstenberg-Sarkozy theorem without using Fourier analysis but still obtaining quantitative bounds. This paper obtains quantitative bounds in the _homogeneous case_ of the problem: that is, the case where the polynomials $P_1,\dots,P_k$ are homogeneous of the same degree, which includes many of the most interesting examples. The approach is along broadly similar lines to the approach of Green, Tao and Ziegler, but the argument is much more complicated: it involves delving into the details of Gowers's proof of Szemeredi's theorem and using "local $U_k$ norms" that were introduced by Tao and Ziegler in order to prove a polynomial generalization of the Green-Tao theorem. The simplest previously unknown case is that of arithmetic progressions of length 3 and square common difference -- that is, the case where we are looking for configurations of the form $(a,a+d^2,a+2d^2)$. Here the result of the paper gives a bound of the form $n=\exp\exp(\delta^{-C})$. In fact, the same sort of bound holds in general, but the dependence of $C$ on the polynomials is not given explicitly, though the methods of the paper could probably be used to give an explicit dependence for anybody sufficiently patient to keep track of all the constants in the argument.

Abstract: A counterexample to a strong variant of the polynomial Freiman-Ruzsa conjecture, Discrete Analysis 2017:8, 6 pp. Given a finite set $A$ of integers, define its _sumset_ $A+A$ to be the set $\{x+y:x,y\in A\}$. A central question in additive combinatorics is the following: what can we say about a set $A$ if its sumset is small? A remarkable theorem of Freiman, later given a different and very influential proof by Imre Ruzsa, gives a complete answer to this question, at least qualitatively speaking, when "small" is interpreted as "of size at most $C|A|$" for an absolute constant $C$. An easy observation is that if $A$ is an arithmetic progression, then $|A+A|=2|A|-1$. Another easy observation is that if $|A+A|\leq C|A|$ and $B\subset A$ is a subset with $|B|\geq c|A|$, then $|B+B|\leq c^{-1}C|B|$. A slightly less easy observation is that "higher-dimensional progressions" have small sumsets: for example, a set $A$ of the form $\{a_0+d_1x+d_2y: 0 \leq x < m_1, 0 \leq y < m_2 \}$, which is a 2-dimensional arithmetic progression, has a sumset of size at most $4|A|$, roughly speaking because it expands by a factor 2 in each direction. Freiman's theorem is the much less obvious statement that these observations give all examples: every set with a small sumset is a large subset of a low-dimensional arithmetic progression. Freiman's original argument gave very weak bounds. Although these were substantially improved by Ruzsa, and improved further by Chang, the theorem nevertheless had the following drawback. If you start with a set $A$ with a sumset of size at most $C|A|$ and apply Freiman's theorem, then you obtain the information that $A$ is contained in an arithmetic progression of size at most $K|A|$ and dimension at most $d$, for constants $K$ and $d$ that depend on $C$. But if you use just this information and go back in the other direction, then all you can deduce is that such a set has sumset of size at most $2^dK|A|$, and even with Chang's bounds for $d$ and $K$, $2^dK$ is far bigger than $C$. Thus, there is a sense in which Freiman's theorem does not capture the full structure of a set $A$ with small sumset. This situation was radically improved in 2010 by Sanders, who obtained a quasipolynomial version of the theorem: that is, if you apply the theorem and its trivial converse, you recover a constant that is at most $\exp((\log C)^t)$ for an absolute constant $t$. However, the holy grail is to obtain a characterization of sets with small sumset (or, as they are often called, sets with small doubling), that worsens the constant by at most a polynomial. The polynomial Freiman-Ruzsa conjecture would, if true, give a characterization of this kind. To state it, we need the notion of a _symmetric convex lattice set_: that is, the intersection of a lattice with a symmetric convex body. If a convex lattice set has dimension $d$, it can be shown that its doubling constant is at most $5^d$. One can project a convex lattice set into a lower dimension and preserve this property, which gives a class of sets of small doubling that is more general than multidimensional arithmetic progressions: a multidimensional arithmetic progression is a projected convex lattice set for which the lattice is $\mathbb Z^d$ and the convex body intersecting it is an axis-aligned cuboid. If $B$ is a projected convex lattice set of bounded dimension, and $X$ is a set of bounded size, then $B+X$ still has small doubling, and the polynomial Freiman-Ruzsa conjecture is roughly that every set $A$ with $|A+A|\leq C|A|$ is contained in such a set, with the sizes of $B$ and $X$ and the dimension of the lattice being such that from this information alone one can recover a doubling constant of $P(C)$ for some fixed polynomial $P$. Above, we observed that projections of convex lattice sets give us a more general class of sets of small doubling than multidimensional arithmetic progressions. But do we need this extra generality, given that we can also pass to large subsets? In a qualitative sense, we do not, by Freiman's theorem (or by a direct argument using Minkowski's second theorem). However, if we want a characterization that gives only a polynomial loss, then it is not obvious whether the extra generality is needed. That is the question addressed by this paper. (The paper deals with subsets of $\mathbb R^n$ rather than sets of integers but this makes no difference.) The authors prove that if one strengthens the polynomial Freiman-Ruzsa conjecture by insisting that the projected lattice convex bodies are multidimensional arithmetic progressions, then the resulting statement is false. This answers a question of Ben Green. In some ways, the result is not surprising: Green expected the strengthened statement to be false, and if it is to be false, then the natural place to look for counterexamples is convex lattice sets that are as dissimilar as possible to aligned cuboids. So the authors' counterexample -- the intersection of a random lattice with a Euclidean ball -- is a natural one. However, it is not clear how to prove that examples of this kind really are counterexamples. The main achievement of this paper is to observe that a recent result (a "reverse Minkowski theorem" due to the second author and Noah Stephens-Davidowitz) can be used to give a short proof, thereby confirming that the statement of the polynomial Freiman-Ruzsa conjecture is the "correct" one.

Abstract: We study the multiparty communication complexity of high dimensional permutations, in the Number On the Forehead (NOF) model. This model is due to Chandra, Furst and Lipton (CFL) who also gave a nontrivial protocol for the Exactly-n problem where three players receive integer inputs and need to decide if their inputs sum to a given integer $n$. There is a considerable body of literature dealing with the same problem, where $(\mathbb{N},+)$ is replaced by some other abelian group. Our work can be viewed as a far-reaching extension of this line of work. We show that the known lower bounds for that group-theoretic problem apply to all high dimensional permutations. We introduce new proof techniques that appeal to recent advances in Additive Combinatorics and Ramsey theory. We reveal new and unexpected connections between the NOF communication complexity of high dimensional permutations and a variety of well known and thoroughly studied problems in combinatorics. Previous protocols for Exactly-n all rely on the construction of large sets of integers without a 3-term arithmetic progression. No direct algorithmic protocol was previously known for the problem, and we provide the first such algorithm. This suggests new ways to significantly improve the CFL protocol. Many new open questions are presented throughout.

Abstract: In this paper, we look at long arithmetic progressions on conics. By an arithmetic progression on a curve, we mean the existence of rational points on the curve whose x-coordinates are in arithmetic progression. We revisit arithmetic progressions on the unit circle, constructing 3-term progressions of points in the first quadrant containing an arbitrary rational point on the unit circle. We also provide infinite families of three term progressions on the unit hyperbola, as well as conics ax2 + cy2 = 1 containing arithmetic progressions as long as 8 terms.

Abstract: Given a real number $0.a_1a_2 a_3\dots$ that is normal to base $b$, we examine increasing sequences $n_i$ so that the number $0.a_{n_1}a_{n_2}a_{n_3}\dots$ are normal to base $b$. Classically it is known that if the $n_i$ form an arithmetic progression then this will work. We give several more constructions, including $n_i$ that are recursively defined based on the digits $a_i$. Of particular interest, we show that if a number is normal to base $b$, then removing all the digits from its expansion which equal $(b-1)$ leaves a base-$(b-1)$ expansion that is normal to base $(b-1)$.

Abstract: Let H be a numerical semigroup. We give effective bounds for the multiplicity e(H) when the associated graded ring grmK[H] is defined by quadrics. We classify Koszul complete intersection semigroups in terms of gluings. Furthermore, for several classes of numerical semigroups considered in the literature (arithmetic, compound, special almost-complete intersections, 3-semigroups, symmetric or pseudosymmetric 4-semigroups) we classify those which are Koszul.

Abstract: An integer sequence is said to be 3-free if no three elements form an arithmetic progression. A Stanley sequence{an} is a 3-free sequence constructed by the greedy algorithm. Namely, given initial terms a0an1 is chosen to be the smallest such that the 3-free condition is not violated. Odlyzko and Stanley conjectured that Stanley sequences divide into two classes based on asymptotic growth: Type 1 sequences satisfy an=(nlog23) and appear well-structured, while Type 2 sequences satisfy an=(n2/logn) and appear disorderly. In this paper, we define the notion of regularity, which is based on local structure and implies Type 1 asymptotic growth. We conjecture that the reverse implication holds. We construct many classes of regular Stanley sequences, which include all known Type 1 sequences as special cases. We show how two regular sequences may be combined into another regular sequence, and how parts of a Stanley sequence may be translated while preserving regularity. Finally, we demonstrate the surprising fact that certain Stanley sequences possess proper subsets that are also Stanley sequences.

Abstract: In this paper, we consider arithmetic progressions contained in Lucas sequences of first and second kind. We prove that for almost all sequences, there are only finitely many and their number can be effectively bounded. We also show that there are only a few sequences which contain infinitely many and one can explicitly list both the sequences and the progressions in them. A more precise statement is given for sequences with dominant root.

Abstract: In an underdetermined system of equations Aw = y, where A is an m × n matrix, only u of the entries of y with u < m are known. Thus Ejw, called 'measurements', are known for certain j ∈ J ⊂ {0, 1, . . . , m - 1} where {Ei, i = 0, 1, . . . , m - 1} are the rows of A and |J| = u. It is required, if possible, to solve the system uniquely when x has at most t non-zero entries with u ≥ 2t. Here such systems are considered from an error-correcting coding point of view. This reduces the problem to finding a suitable decoding algorithm which then finds w. Decoding workable algorithms are shown to exist, from which the unknown w may be determined, in cases where the known 2t values are evenly spaced, that is when the elements of J are in arithmetic sequence. The method can be applied to Fourier matrices and certain classes of Vandermonde matrices. The decoding algorithm has complexity O(nt) and in some cases the complexity is O(t2). Matrices which have the property that the determinant of any square submatrix is non-zero are of particular interest. Randomly choosing rows of such matrices can then give t error-correcting pairs to generate a 'measuring' code. This has applications to signal processing and compressed sensing.

Abstract: The high peak value of the transmission signal of wireless communication systems lead to wasteful energy consumption and degradation of several transmission performances. We continue the theoretical contributions made in [1], [2] towards the understanding of tone reservation method for orthogonal transmission schemes. There it was shown that the combinatorial object called arithmetic progression plays an important role in setting limitations for the applicability of the tone reservation method for OFDM system. In this work, we introduce the combinatorial object called perfect Walsh sum (PWS), playing a similar role for CDMA systems as arithmetic progression for OFDM systems. We show that for a given m, n ∊ N and δ ∊ (0, 1), every subset I of the set [N] of the first N=2n numbers, which fulfills |I|/N ≥ δ and |I| ≥ 2(2/δ)2m − 1, contains a PWS of size 2m. Consequences of the latter are results analogous to the famous Szemeredi Theorem on arithmetic progressions, Conlon-Gower's Theorem on probabilistic construction of “sparse” sets containing an arithmetic progression, and even a solution of Erdos' conjecture on arithmetic progressions. Those results give in particular an insight into the asymptotic behaviour of tone reservation method for CDMA systems.

Abstract: In this paper, we present several explicit formulas of the sums and hyper-sums of the powers of the first (n+1)-terms of a general arithmetic sequence in terms of Stirling numbers and generalized Bernoulli polynomials.

Abstract: Improving upon previous work [3] on the subject, we use Wright’s Circle Method to derive an asymptotic formula for the number of parts in all partitions of an integer n that are in any given arithmetic progression.

Abstract: Let $\theta(n)$ denote the number of permutations of $\{1,2,\ldots,n\}$ that do not contain a 3-term arithmetic progression as a subsequence. Such permutations are known as 3-free permutations. We present a dynamic programming algorithm to count all 3-free permutations of $\{1,2,\ldots,n\}$. We use the output to extend and correct enumerative results in the literature for $\theta(n)$ from $n=20$ out to $n=90$ and use the new values to inductively improve existing bounds on $\theta(n)$.

Abstract: A sum-and-distance system is a collection of finite sets of integers such that the sums and differences formed by taking one element from each set generate a prescribed arithmetic progression. Such systems, with two component sets, arise naturally in the study of matrices with symmetry properties and consecutive integer entries. Sum systems are an analogous concept where only sums of elements are considered. We establish a bijection between sum systems and sum-and-distance systems of corresponding size, and show that sum systems are equivalent to principal reversible cuboids, which are tensors with integer entries and a symmetry of "reversible square" type. We prove a structure theorem for principal reversible cuboids, which gives rise to an explicit construction formula for all sum systems in terms of joint ordered factorisations of their component set cardinalities.

Abstract: A trapezoidal number, a sum of at least two consecutive positive integers, is a figurate number that can be represented by points rearranged in the plane as a trapezoid. Such numbers have been of interest and extensively studied. In this paper, a generalization of trapezoidal numbers has been introduced. For each positive integer , a positive integer is called an -trapezoidal number if can be written as an arithmetic series of at least terms with common difference . Properties of -trapezoidal numbers have been studied together with their trapezoidal representations. In the special case where , the characterization and enumeration of such numbers have been given as well as illustrative examples. Precisely, for a fixed -trapezoidal number , the ways and the number of ways to write as an arithmetic series with common difference have been determined. Some remarks on -trapezoidal numbers have been provided as well.

Abstract: If $E$ is an elliptic curve over $\mathbb{Q}$, then it follows from work of Serre and Hooley that, under the assumption of the Generalized Riemann Hypothesis, the density of primes $p$ such that the group of $\mathbb{F}_p$-rational points of the reduced curve $\tilde{E}(\mathbb{F}_p)$ is cyclic can be written as an infinite product $\prod \delta_\ell$ of local factors $\delta_\ell$ reflecting the degree of the $\ell$-torsion fields, multiplied by a factor that corrects for the entanglements between the various torsion fields. We show that this correction factor can be interpreted as a character sum, and the resulting description allows us to easily determine non-vanishing criteria for it. We apply this method in a variety of other settings. Among these, we consider the aforementioned problem with the additional condition that the primes $p$ lie in a given arithmetic progression. We also study the conjectural constants appearing in Koblitz's conjecture, a conjecture which relates to the density of primes $p$ for which the cardinality of the group of $\mathbb{F}_p$-points of $E$ is prime.

Abstract: In this paper, we consider monomial curves in ${\mathbb A}_k^3$ parameterized by $t \rightarrow (t^{2q +1}, t^{2q +1 + m}, t^{2q +1 +2 m})$ where $gcd( 2q+1,m)=1$. The symbolic blowup algebras of these monomial curves is Gorenstein (\cite{goto-nis-shim}, \cite{goto-nis-shim-2}). We give a simple proof for the the Gorenstein property for the symbolic blowup algebras of these curves.

Abstract: Abstract: We define an analogous pair of recurrence relations that yield some Fibonacci, Lucas and generalized Fibonacci identities with the coefficients in arithmetic progression. One relation yields same sign identities and the other alternating signs identities. We also show some new results for negative indexed Fibonacci and Lucas Sequences.

Abstract: 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.

Abstract: Schmidt's game is generally used to deduce qualitative information about the Hausdorff dimensions of fractal sets and their intersections. However, one can also ask about quantitative versions of the properties of winning sets. In this paper we show that such quantitative information has applications to various questions including: * What is the maximal length of an arithmetic progression on the "middle $\epsilon$" Cantor set? * What is the smallest $n$ such that there is some element of the ternary Cantor set whose continued fraction partial quotients are all $\leq n$? * What is the Hausdorff dimension of the set of $\epsilon$-badly approximable numbers on the Cantor set? We show that a variant of Schmidt's game known as the $potential$ $game$ is capable of providing better bounds on the answers to these questions than the classical Schmidt's game. We also use the potential game to provide a new proof of an important lemma in the classical proof of the existence of Hall's Ray.

Abstract: Ovaj rad započinje s dva poznata teorema aditivne kombinatorike, koji su potom svedeni na rezultat iz teorije grafova. Daljnje poopćenje je iskazano jezikom teorije vjerojatnosti, kako bi ga se dokazalo korištenjem vjerojatnosne varijante Szemerédijeve leme o regularnosti. Ta lema daje dekompoziciju proizvoljne slučajne varijable na strukturirani dio, pseudoslučajni dio i grešku, a u radu je iznesen njezin potpuni dokaz.

Abstract: In this paper, we present closed forms for certain finite sums. In each case, the denominator of the summand is a product of sine or cosine functions, where at least one of these functions is squared. Furthermore, in each case, the arguments of the trigonometric functions in the denominator of the summand increase in arithmetic progression.