Infinite Arithmetic Progression Research Articles

On the set of values of a positive ternary quadratic form

It is shown that the set of values of an integer positive definite quadratic form in three variables contains an infinite arithmetic progression.

On Eventually Periodic Sets as Minimal Additive Complements

We say a subset $C$ of an abelian group $G$ arises as a minimal additive complement if there is some other subset $W$ of $G$ such that $C+W=\{c+w:c\in C,\ w\in W\}=G$ and such that there is no proper subset $C'\subset C$ such that $C'+W=G$. In their recent paper, Burcroff and Luntzlara studied, among many other things, the conditions under which eventually periodic sets, which are finite unions of infinite (in the positive direction) arithmetic progressions and singletons, arise as minimal additive complements in $\mathbb Z$. In the present paper we study this further and give, in the form of bounds on the period $m$, some sufficient conditions for an eventually periodic set to arise as a minimal additive complement; in particular we show that "all eventually periodic sets are eventually minimal additive complements''. Moreover, we generalize this to a framework in which "patterns'' of points (subsets of $\mathbb Z^2$) are projected down to $\mathbb Z$, and we show that all sets which arise this way are eventually minimal additive complements. We also introduce a formalism of formal power series, which serves purely as a bookkeeper in writing down proofs, and we prove some basic properties of these series (e.g. sufficient conditions for inverses to be unique). Through our work we are able to answer a question of Burcroff and Luntzlara (when does $C_1\cup(-C_2)$ arise as a minimal additive complement, where $C_1,C_2$ are eventually periodic sets?) in a large class of cases.

Large subsets of Euclidean space avoiding infinite arithmetic progressions

It is known that if a subset of R \mathbb {R} has positive Lebesgue measure, then it contains arbitrarily long finite arithmetic progressions. We prove that this result does not extend to infinite arithmetic progressions in the following sense: for each λ \lambda in [ 0 , 1 ) [0,1) , we construct a subset of R \mathbb {R} that intersects every interval of unit length in a set of measure at least λ \lambda , but that does not contain any infinite arithmetic progression.

On power values of sum of divisors function in arithmetic progressions

Let $$a\ge 1, b\ge 0$$ and $$k\ge 2$$ be any given integers. It has been proven that there exist infinitely many natural numbers m such that sum of divisors of m is a perfect kth power. We try to generalize this result when the values of m belong to any given infinite arithmetic progression $$an+b$$ . We prove if a is relatively prime to b and order of b modulo a is relatively prime to k then there exist infinitely many natural numbers n such that sum of divisors of $$an+b$$ is a perfect kth power. We also prove that, in general, either sum of divisors of $$an+b$$ is not a perfect kth power for any natural number n or sum of divisors of $$an+b$$ is a perfect kth power for infinitely many natural numbers n.

Entropy rate of product of independent processes

We study the multiplicative version of the classical Furstenberg’s filtering problem, where instead of the sum textbf{X}+textbf{Y} one considers the product textbf{X}cdot textbf{Y} (textbf{X} and textbf{Y} are bilateral, real, finitely-valued, stationary independent processes, textbf{Y} is taking values in {0,1}). We provide formulas for {textbf{H}},(textbf{X}cdot textbf{Y},,|,,textbf{Y}). As a consequence, we show that if {textbf{H}},({textbf{X}})>{textbf{H}},({textbf{Y}})=0 and textbf{X}amalg textbf{Y}, then textbf{H},(textbf{X}cdot textbf{Y})<{textbf{H}},({textbf{X}}) (and thus textbf{X} cannot be filtered out from textbf{X}cdot textbf{Y}) whenever textbf{X} is not bilaterally deterministic, textbf{Y} is ergodic and textbf{Y} first return to 1 can take arbitrarily long with positive probability. On the other hand, if almost surely textbf{Y} visits 1 along an infinite arithmetic progression of a fixed difference (with possibly some more visits in between) then we can find textbf{X} that is not bilaterally deterministic and such that {textbf{H}},(textbf{X}cdot textbf{Y})={textbf{H}},({textbf{X}}). As a consequence, a {mathscr {B}}-free system (X_eta ,S) is proximal if and only if there is always an entropy drop h(kappa *nu _eta )<h(kappa ) for any kappa corresponding to a non-bilaterally deterministic process of positive entropy. These results partly settle some open problems on invariant measures for {mathscr {B}}-free systems.

On arithmetic progressions in non-periodic self-affine tilings

AbstractWe study the repetition of patches in self-affine tilings in ${\mathbb {R}}^d$ . In particular, we study the existence and non-existence of arithmetic progressions. We first show that an arithmetic condition of the expansion map for a self-affine tiling implies the non-existence of certain one-dimensional arithmetic progressions. Next, we show that the existence of full-rank infinite arithmetic progressions, pure discrete dynamical spectrum, and limit-periodicity are all equivalent for a certain class of self-affine tilings. We finish by giving a complete picture for the existence or non-existence of full-rank infinite arithmetic progressions in the self-similar tilings in ${\mathbb {R}}^d$ .

Littlewood polynomials, spectral-null codes, and equipowerful partitions

Let $[n]$ denote $\{0,1, ... , n-1\}$. A polynomial $f(x) = \sum a_i x^i$ is a Littlewood polynomial (LP) of length $n$ if the $a_i$ are $\pm 1$ for $i \in [n]$, and $a_i = 0$ for $i \ge n$. Such an LP is said to have order $m$ if it is divisible by $(x-1)^m$. The problem of finding the set $L_m$ of lengths of LPs of order $m$ is equivalent to finding the lengths of spectral-null codes of order $m$, and to finding $n$ such that $[n]$ admits a partition into two subsets whose first $m$ moments are equal. Extending the techniques and results of Boyd and others, we completely determine $L_7$ and $L_8$ and prove that 192 is the smallest element of $L_9$. Our primary tools are the use of carefully targeted searches using integer linear programming (both to find LPs and to disprove their existence for specific $n$ and $m$), and an unexpected new concept (that arose out of observed symmetry properties of LPs) that we call regenerative pairs, which produce infinite arithmetic progressions in $L_m$. We prove that for $m \le$ 8, whenever there is an LP of length $n$ and order $m$, there is one of length $n$ and order $m$ that is symmetric (resp.~antisymmetric) if m is even (resp.~odd).

Decimation and interleaving operations in one-sided symbolic dynamics

This paper studies subsets of one-sided shift spaces on a finite alphabet. Such subsets arise in symbolic dynamics, in fractal constructions, and in number theory. We study a family of decimation operations, which extract subsequences of symbol sequences in infinite arithmetic progressions, and show these operations are closed under composition. We also study a family of n-ary interleaving operations, one for each n≥1. Given subsets X0,X1,...,Xn−1 of such a shift space, the n-ary interleaving operation produces a set whose elements combine individual elements xi, one from each Xi, by interleaving their symbol sequences in arithmetic progressions (modn). We determine algebraic relations between decimation and interleaving operations and the shift map. We study set-theoretic n-fold closure operations X↦X[n], which interleave decimations of X of modulus level n. A set is n-factorizable if X=X[n]. The n-fold interleaving operations are closed under composition and are idempotent. To each X we assign the set N(X) of all values n≥1 for which X=X[n]. We characterize the possible sets N(X) as nonempty sets of positive integers that form a distributive lattice under the divisibility partial order and are downward closed under divisibility. We show that all sets of this type occur. We introduce a class of weakly shift-stable sets and show that this class is closed under all decimation, interleaving, and shift operations. We study two notions of entropy for subsets of the full one-sided shift and show that they coincide for weakly shift-stable X, but can be different in general. We give a formula for entropy of interleavings of weakly shift-stable sets in terms of individual entropies.

On a problem of Erdős, Nathanson and Sárközy

TextIn 1988, Erdős, Nathanson and Sárközy proved that, if A is a set of nonnegative integers with the lower asymptotic density 1/k, where k is a positive integer, then (k+1)A must contain an infinite arithmetic progression with difference at most k2−k, where (k+1)A is the set of all sums of k+1 elements of A, and asked if (k+1)A must contain an infinite arithmetic progression with difference at most O(k). In this paper, we solve this problem by proving that, for every sufficiently large integer k, there exists a set A of nonnegative integers with the lower asymptotic density 1/k such that (k+1)A does not contain an infinite arithmetic progression with difference less than k1.5. We also study the similar problem when A has the lower asymptotic density 0<w<1. VideoFor a video summary of this paper, please visit https://youtu.be/B1QJc2IFSIw.

On square-free arithmetic progressions in infinite words

Let w be an infinite word and p≥2 a given positive integer. Denote by [w]p the infinite arithmetic progression modulo p of w, i.e., it is the substring of w consisting of the concatenation of the letters at positions kp for k=1,2,… We show that in the ternary case, for each p≥3 there exists an infinite square-free word w such that also the arithmetic progression [w]p is square-free. For the case p=2 there are no solutions.

ON THE ABSENCE OF ZEROS IN INFINITE ARITHMETIC PROGRESSION FOR CERTAIN ZETA FUNCTIONS

Putnam [‘On the non-periodicity of the zeros of the Riemann zeta-function’, Amer. J. Math.76 (1954), 97–99] proved that the sequence of consecutive positive zeros of $\unicode[STIX]{x1D701}(\frac{1}{2}+it)$ does not contain any infinite arithmetic progression. We extend this result to a certain class of zeta functions.

Intervals without primes near elements of linear recurrence sequences

Let [Formula: see text] be an unbounded sequence of integers satisfying a linear recurrence relation with integer coefficients. We show that for any [Formula: see text] there exist infinitely many [Formula: see text] for which [Formula: see text] consecutive integers [Formula: see text] are all divisible by certain primes. Moreover, if the sequence of integers [Formula: see text] satisfying a linear recurrence relation is unbounded and non-degenerate then for some constant [Formula: see text] the intervals [Formula: see text] do not contain prime numbers for infinitely many [Formula: see text]. Applying this argument to sequences of integer parts of powers of Pisot and Salem numbers [Formula: see text] we derive a similar result for those sequences as well which implies, for instance, that the shifted integer parts [Formula: see text], where [Formula: see text] and [Formula: see text] runs through some infinite arithmetic progression of positive integers, are all composite.

A Skolem–Mahler–Lech Theorem for Iterated Automorphisms ofK–algebras

AbstractIn this paper we prove a commutative algebraic extension of a generalized Skolem–Mahler– Lech theorem. LetAbe a finitely generated commutativeK–algebra over a field of characteristic 0, and letσbe aK–algebra automorphism ofA. Given idealsIandJofA, we show that the setSof integersmsuch thatJis a finite union of complete doubly infinite arithmetic progressions inm, up to the addition of a finite set. Alternatively, this result states that for an affine schemeXof finite type overK, an automorphismσ∊ 2 AutK(X), andYandZany two closed subschemes ofX, the set of integersmwithYis as above. We present examples showing that this result may fail to hold if the affine schemeXis not of finite type, or ifXis of finite type but the fieldKhas positive characteristic.

Number theoretic applications of a class of Cantor series fractal functions, II

It is well known that all numbers that are normal of order k in base b are also normal of all orders less than k. Another basic fact is that every real number is normal in base b if and only if it is simply normal in base bkfor all k. This may be interpreted to mean that a number is normal in base b if and only if all blocks of digits occur with the desired relative frequency along every infinite arithmetic progression. We reinterpret these theorems for the Q-Cantor series expansions and show that they are no longer true in a particularly strong way. The main theoretical result of this paper will be to reduce the problem of constructing normal numbers with certain pathological properties to the problem of solving a system of Diophantine relations.

Covering an arithmetic progression with geometric progressions and vice versa

We show that there exists a positive constant C such that the following holds: Given an infinite arithmetic progression [Formula: see text] of real numbers and a sufficiently large integer n (depending on [Formula: see text]), there is a need of at least Cn geometric progressions to cover the first n terms of [Formula: see text]. A similar result is presented, with the role of arithmetic and geometric progressions reversed.

On the distribution of orbits in affine varieties

Given an affine variety $X$, a morphism ${\it\phi}:X\rightarrow X$, a point ${\it\alpha}\in X$, and a Zariski-closed subset $V$ of $X$, we show that the forward ${\it\phi}$-orbit of ${\it\alpha}$ meets $V$ in at most finitely many infinite arithmetic progressions, and the remaining points lie in a set of Banach density zero. This may be viewed as a weak asymptotic version of the dynamical Mordell–Lang conjecture for affine varieties. The results hold in arbitrary characteristic, and the proof uses methods of ergodic theory applied to compact Berkovich spaces.

Finite and Infinite Arithmetic Progressions Related to Beta-Expansion

Let1&lt;β&lt;2andε(x,β)be theβ-expansion ofx∈[0,1). Denote byAβ(x)the set of positions where the digit 1 appears inε(x,β). We consider the sets of pointsxsuch thatAβ(x)contains arbitrarily long arithmetic progressions and includes infinite arithmetic progressions, respectively. Their sizes are investigated from the topological, metric, and dimensional viewpoints.

Monadic structures over an ordered universal random graph and finite automata

We continue the investigation of the expressive power of the language of predicate logic for finite algebraic systems embedded in infinite systems. This investigation stems from papers of M. A. Taitslin, M. Benedikt and L. Libkin, among others. We study the properties of a finite monadic system which can be expressed by formulae if such a system is embedded in a random graph that is totally ordered in an arbitrary way. The Büchi representation is used to connect monadic structures and formal languages. It is shown that, if one restricts attention to formulae that are -invariant in totally ordered random graphs, then these formulae correspond to finite automata. We show that -invariant formulae expressing the properties of the embedded system itself can express only Boolean combinations of properties of the form `the cardinality of an intersection of one-place predicates belongs to one of finitely many fixed finite or infinite arithmetic progressions'.

On the intersection of infinite geometric and arithmetic progressions

We prove that the intersection G ∩ A of an infinite geometric progression G = u, uq, uq2, uq3, ..., where u > 0 and q > 1 are real numbers, and an infinite arithmetic progression A contains at most 3 elements except for two kinds of ratios q. The first exception occurs for q = r1/d, where r > 1 is a rational number and d ∈ ℕ. Then this intersection can be of any cardinality s ∈ ℕ or infinite. The other (possible) exception may occur for q = β1/d, where β > 1 is a real cubic algebraic number with two nonreal conjugates of moduli distinct from β and d ∈ ℕ. In this (cubic) case, we prove that the intersection G ∩ A contains at most 6 elements.

A gap principle for dynamics

AbstractLet f1,…,fg∈ℂ(z) be rational functions, let Φ=(f1,…,fg) denote their coordinate-wise action on (ℙ1)g, let V ⊂(ℙ1)g be a proper subvariety, and let P be a point in (ℙ1)g(ℂ). We show that if 𝒮={n≥0:Φn(P)∈V (ℂ)} does not contain any infinite arithmetic progressions, then 𝒮 must be a very sparse set of integers. In particular, for any k and any sufficiently large N, the number of n≤N such that Φn(P)∈V (ℂ) is less than log kN, where log k denotes the kth iterate of the log function. This result can be interpreted as an analogue of the gap principle of Davenport–Roth and Mumford.