Given two nonempty subsets $A, B$ of a group $G$, they are said to form a co-minimal pair if $A \cdot B = G$, and $A' \cdot B \subsetneq G$ for any $\emptyset \neq A' \subsetneq A$ and $A\cdot B' \subsetneq G$ for any $\emptyset \neq B' \subsetneq B$. In this article, we show several new results on co-minimal pairs in the integers and the integral lattices. We prove that for any $d\geq 1$, the group $\mathbb{Z}^{2d}$ admits infinitely many automorphisms such that for each such automorphism $σ$, there exists a subset $A$ of $\mathbb{Z}^{2d}$ such that $A$ and $σ(A)$ form a co-minimal pair. The existence and construction of co-minimal pairs in the integers with both the subsets $A$ and $B$ ($A\neq B$) of infinite cardinality was unknown. We show that such pairs exist and explicitly construct these pairs satisfying a number of algebraic properties.
Let $C,W\subseteq \mathbb{Z}$. If $C+W=\mathbb{Z}$, then the set $C$ is called an additive complement to $W$ in $\mathbb{Z}$. If no proper subset of $C$ is an additive complement to $W$, then $C$ is called a minimal additive complement. Let $X\subseteq \mathbb{N}$. If there exists a positive integer $T$ such that $x+T\in X$ for all sufficiently large integers $x\in X$, then we call $X$ eventually periodic. In this paper, we study the existence of a minimal complement to $W$ when $W$ is eventually periodic or not. This partially answers a problem of Nathanson.
A subset $C$ of an abelian group $G$ is a minimal additive complement to $W \subseteq G$ if $C + W = G$ and if $C' + W \neq G$ for any proper subset $C' \subset C$. In this paper, we study which sets of integers arise as minimal additive complements. We confirm a conjecture of Kwon, showing that bounded-below sets with arbitrarily large gaps arise as minimal additive complements. Moreover, our construction shows that any such set belongs to a co-minimal pair, strengthening a result of Biswas and Saha for lacunary sequences. We bound the upper and lower Banach density of syndetic sets that aris…
We consider the integer Chebyshev problem, that of minimizing the supremum norm over polynomials with integer coefficients on the interval $[0,1]$. We implement algorithms from semi-infinite programming and a branch and bound algorithm to improve on previous methods for finding integer Chebyshev polynomials of degree $n$. Using our new method, we found 16 new integer Chebyshev polynomials of degrees in the range 147 to 244.
In this article, the goal is to find the shortest presentation of a matrix $A \in PSL(2,\mathbb{Z})$ in terms of the so-called continuant matrices which are most known for their role in continued fraction theory. In chapter 7 of arXiv:1811.01229, Morier-Génoud and Ovsienko investigate this problem with the restriction that all coefficients of the continuant matrices are positive. Now, the goal is to determine the shortest presentation allowing all integer coefficients. To determine this minimal presentation, a few characteristic transformations will be introduced. It will also be investigated …
A set of integers $S \subset \mathbb{N}$ is an $α$-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on $α$, more specifically if $| (x+w) - (y+z) | \geq \max \{ x^α,y^α,z^α,w^α\}$ for every $x,y,z,w \in S$ satisfying $\max \{x,w\} \neq \max \{y,z\}$. We obtain a new lower bound for the growth of $α$-strong infinite Sidon sets when $0 \leq α< 1$. We also further extend that notion in a natural way by obtaining the first non-trivial bound for $α$-strong infinite $B_h$ sets. In both cases, we study the implications of these bounds for the density …
A set $A$ of nonnegative integers is an asymptotic basis of order $h$ if every sufficiently large integer can be represented as the sum of $h$ not necessarily distinct elements of $A$. The asymptotic basis $A$ is minimal if removing any element of $A$ destroys every representation of infinitely many integers, and so $A\setminus \{a\}$ is not an asymptotic basis of order $h$ for all $a\in A$. In this paper, a new class of minimal asymptotic bases is constructed.
We prove that the gcd of certain infinite number of integers associated to generalised arithmetic progressions remains bounded independent of the progression. Using this we also get bounds on the indices of certain congruence subgroups of arithmetic groups.
If p is a prime and n a positive integer, let v(n) denote the exponent of p in n, and u(n)=n/p^{v(n)} the unit part of n. If k is a positive integer not divisible by p, we show that the p-adic limit of (-1)^{pke} u((kp^e)!) as e goes to infinity is a well-defined p-adic integer, which we call z_k. In terms of these, we give a formula for the p-adic limit of binom{a p^e +c, b p^e +d) as e goes to infinity, which we call binom(a p^\infty +c, b p^\infty +d). Here a \ge b are positive integers, and c and d are integers.
In this paper, we use some extension of the Cayley-Hamilton theorem to find a family of matrices with integer entries that satisfy the non-linear Diophantine equation $ x^{n}+y^{p}=z^{q}$ where $n,p$ and $q$ are arbitrary positive integers.
In 1949, Motzkin proved that every Euclidean domain $R$ has a minimal Euclidean function, $ϕ_R$. He showed that when $R = \mathbb{Z}$, the minimal function is $ϕ_{\mathbb{Z}}(x) = \lfloor \log_2 |x| \rfloor$. For over seventy years, $ϕ_{\mathbb{Z}}$ has been the only example of an explictly-computed minimal function in a number field. We give the first explicitly-computed minimal function in a non-trivial number field, $ϕ_{\mathbb{Z}[i]}$, which computes the length of the shortest possible $(1+i)$-ary expansion of any Gaussian integer. We also present an algorithm that uses $ϕ_{\mathbb{Z}[i]}$…
Let $W\subseteq \mathbb{Z}$ be a non-empty subset of the integers. A nonempty set $C\subseteq \mathbb{Z}$ is said to be an asymptotic complement to $W$ if $W+C$ contains almost all the integers except a set of finite size. $C$ is said to be a minimal asymptotic complement if $C$ is an asymptotic complement, but $C\setminus \lbrace c\rbrace$ is not an asymptotic complement $\forall c\in C$. Asymptotic complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others, while the notion of minimal asymptotic complements is due to Nathan…
Parallel addition, i.e., addition with limited carry propagation, has been so far studied for complex bases and integer alphabets. We focus on alphabets consisting of integer combinations of powers of the base. We give necessary conditions on the alphabet allowing parallel addition. Under certain assumptions, we prove the same lower bound on the size of the generalized alphabet that is known for alphabets consisting of consecutive integers. We also extend the characterization of bases allowing parallel addition to numeration systems with non-integer alphabets.
We present a versatile construction allowing one to obtain pairs of integer sets with infinite symmetric difference, infinite intersection, and identical representation functions.
The minimal excludant of a partition $λ$, $\rm{mex}(λ)$, is the smallest positive integer that is not a part of $λ$. For a positive integer $n$, $ σ\, \rm{mex}(n)$ denotes the sum of the minimal excludants of all partitions of $n$. Recently, Andrews and Newman obtained a new combinatorial interpretations for $σ\, \rm{mex}(n)$. They showed, using generating functions, that $σ\, \rm{mex}(n)$ equals the number of partitions of $n$ into distinct parts using two colors. In this paper, we provide a purely combinatorial proof of this result and new properties of the function $σ\, \rm{mex}(n)$. We gen…
A group of infinite products over low-order rational polynomials evaluated at the sequence of prime numbers is loosely called the Hardy-Littlewood constants. In this manuscript we look at them as factors embedded in a super-product over primes, semiprimes, 3-almost primes etc. Numerical tables are derived by transformation into series over k-almost prime zeta functions. Alternative product representations in a basis of k-almost prime products associated with Euler's formula for the Riemann zeta function are also pointed out.
Let O be the ring of S-integers in a number field k. We prove that if the group of units O^* is infinite then every matrix in $Γ$ = SL_2(O) is a product of at most 9 elementary matrices. This completes a long line of research in this direction. As a consequence, we obtain that $Γ$ is boundedly generated as an abstract group.
For any non-negative integer $n$ and non-zero integer $r$, let $p_r(n)$ denote Ramanujan's general partition function. By employing $q$-identities, we prove some new Ramanujan-type congruences modulo 5 for $p_r(n)$ for $r=-(5λ+1), -(5λ+3), -(5λ+4), -(25λ+1)$, $-(25λ+2)$, and any integer $λ$.
We determine the minimal non-trivial integer group determinant for the dicyclic group of order $4n$ when $n$ is odd. We also discuss the set of all integer group determinants for the dicyclic groups of order $4p$.
Let $A$ be an infinite set of nonnegative integers. For $h \geq 2$, let $hA$ be the set of all sums of $h$ not necessarily distinct elements of $A$. If every sufficiently large integer in the sumset $hA$ has at least two representations, then $A(x) \geq (\log x)/\log h)-w_0$, where $A(x)$ counts the number of integers $a \in A$ such that $1 \leq a \leq x$.
We present a practical algorithm to compute models of rational functions with minimal resultant under conjugation by fractional linear transformations. We also report on a search for rational functions of degrees 2 and 3 with rational coefficients that have many integers in a single orbit. We find several minimal quadratic rational functions with 8 integers in an orbit and several minimal cubic rational functions with 10 integers in an orbit. We also make some elementary observations on possibilities of an analogue of Szpiro's conjecture in a dynamical setting and on the structure of the set o…
We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidability. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. In particular, we show that the following propositions hold. (1) For any rational prime $q$ and any positive rational integer $m$, algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by $q^{m}$. (…
The study of minimal complements in a group or a semigroup was initiated by Nathanson. The notion of minimal complements and being a minimal complement leads to the notion of co-minimal pairs which was considered in a prior work of the authors. In this article, we study which type of subsets in the integers and free abelian groups of higher rank can be a part of a co-minimal pair. We show that a majority of lacunary sequences have this property. From the conditions established, one can show that any infinite subset of any finitely generated abelian group has uncountably many subsets which is a…
We provide a lower bound for the dimension of the vector space spanned by 1 and by the values of the Riemann Zeta function at the first odd integers. As a consequence, the Zeta function takes infinitely many irrational values at odd integers.
In this paper, we use a simple discrete dynamical model to study integer partitions and their lattice. The set of reachable configurations of the model, with the order induced by the transition rule defined on it, is the lattice of all partitions of an integer, equipped with a dominance ordering. We first explain how this lattice can be constructed by an algorithm in linear time with respect to its size by showing that it has a self-similar structure. Then, we define a natural extension of the model to infinity, which we compare with the Young lattice. Using a self-similar tree, we obtain an e…
Every Euclidean domain $R$ has a minimal Euclidean function, $ϕ_R$. A companion paper \cite{Graves} introduced a formula to compute $ϕ_{\mathbb{Z}[i]}$. It is the first formula for a minimal Euclidean function for the ring of integers of a non-trivial number field. It did so by studying the geometry of the set $B_n = \left \{ \sum_{j=0}^n v_j (1+i)^j : v_j \in \{0, \pm 1, \pm i \} \right \}$ and then applied Lenstra's result that $ϕ_{\mathbb{Z}[i]}^{-1}([0,n]) = B_n$ to provide a short proof of $ϕ_{\mathbb{Z}[i]}$. Lenstra's proof requires s substantial algebra background. This paper uses the …
We study the dynamic structures of the monomial $x^m$ over the ring of $p$-adic integers for every positive integer $m$ and for primes $p=2,3$ and $5$. The dynamic structures are described by investigating minimal decompositions which consist of minimal subsystems and attracting basins.
The number theoretic analogue of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g > 1, the study of h-nets in the additive group of integers with respect to the generating set A_g = {g^i:i=0,1,2,...} requires a knowledge of the word lengths of integers with respect to A_g. A g-adic representation of an integer is described that algorithmically produces a representation of shortest length. Additive complements and additive asymptotic complements are also discussed, together with their associated minimality…
There are scattered results in the literature showing that the leading principal minors of certain infinite integer matrices form the Fibonacci and Lucas sequences. In this article, among other results, we have obtained new families of infinite matrices such that the leading principal minors of them form a famous integer (sub)sequence, such as Fibonacci, Lucas, Pell and Jacobsthal (sub)sequences.
It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric familiy of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $ξ$ of the polynomial $P_t(x)=x^4-tx^3-6x^2+tx+1$, assuming that $t>0$, $t\neq 3$ and $t^2+16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal index and all elements of minimal index in …
Two infinite sets $A$ and $B$ of nonnegative integers are called additive complements if their sumset contains every nonnegative integer. In 1964, Danzer constructed infinite additive complements $A$ and $B$ with $A(x)B(x) = (1 + o(1))x$ as $x \rightarrow \infty$, where $A(x)$ and $B(x)$ denote the counting function of the sets $A$ and $B$, respectively. In this paper we solve a problem of Chen and Fang by extending the construction of Danzer.
Inspired by the Erdös-Turan conjecture we consider subsets of the natural numbers that contains infinitely many aritmetic progressions (APs) of any given length - such sets will be called AP-sets and we know due to the Green-Tao Theorem and Sźmeredis Theorem that the primes and all subsets of positive upper density are AP-sets. We prove that (1, 1,..., 1) is a solution to the equation Mx = 0 where M is an integer matrix whose null space has dimension at least 2, if and only if the equation has infinitely many solutions such that the coordinates of each solution are elements in the same AP. Thi…
By a sphere-packing argument, we show that there are infinitely many pairs of primes that are close to each other for some metrics on the integers. In particular, for any numeration basis $q$, we show that there are infinitely many pairs of primes the base $q$ expansion of which differ in at most two digits. Likewise, for any fixed integer $t,$ there are infinitely many pairs of primes, the first $t$ digits of which are the same. In another direction, we show that, there is a constant $c$ depending on $q$ such that for infinitely many integers $m$ there are at least $c\log \log m$ primes which…
Let $(m_1,\ldots,m_J)$ and $(r_1,\ldots,r_J)$ be two sequences of $J$ positive integers satisfying $1\le r_j< m_j$ for all $j=1,\ldots,J$. Let $(δ_1,\ldots,δ_J)$ be a sequence of $J$ nonzero integers. In this paper, we study the asymptotic behavior of the Taylor coefficients of the infinite product $$\prod_{j=1}^J\Bigg(\prod_{k\ge 1}\big(1-q^{r_j+m_j(k-1)}\big)\big(1-q^{-r_j+m_jk}\big)\Bigg)^{δ_j}.$$
Given a finite set of bases $b_1$, $b_2$, \dots, $b_r$ (integers greater than $1$), a multi-base representation of an integer~$n$ is a sum with summands $db_1^{α_1}b_2^{α_2} \cdots b_r^{α_r}$, where the $α_j$ are nonnegative integers and the digits $d$ are taken from a fixed finite set. We consider multi-base representations with at least two bases that are multiplicatively independent. Our main result states that the order of magnitude of the minimal Hamming weight of an integer~$n$, i.e., the minimal number of nonzero summands in a representation of~$n$, is $\log n / (\log \log n)$. This is …
For each prime number $p$, the dynamical behavior of the square mapping on the ring $\mathbb{Z}_p$ of $p$-adic integers is studied. For $p=2$, there are only attracting fixed points with their attracting basins. For $p\geq 3$, there are a fixed point $0$ with its attracting basin, finitely many periodic points around which there are countably many minimal components and some balls of radius $1/p$ being attracting basins. All these minimal components are precisely exhibited for different primes $p$.
We investigate class field towers of number fields obtained as fixed fields of modular representations of the absolute Galois group of the rational numbers. First, for each $k\in\{12,16,18,20,22,26\}$, we give explicit rational primes $ł$ such that the fixed field of the mod-$ł$ representation attached to the unique normalized cusp eigenforms of weight $k$ on $\Sl_2(\Z)$ has an infinite class field tower. Under a conjecture of Hardy and Littlewood, we further prove that there exist infinitely many such primes for each $k$ (in the above list). Second, given a non-CM curve $E/\Q$, we show that t…
Using a quartic surface and its rational curves we can give an infinite number of integer hexahedra; these are 6 sided 3d solids, each face a trapezoid, with all sides and diagonals having intger lengths.
Let H(N) denote the set of all polynomials with positive integer coefficients which have their zeros in the open left half-plane. We are looking for polynomials in H(N) whose largest coefficients are as small as possible and also for polynomials in H(N) with minimal sum of the coefficients. Let h(N) and s(N) denote these minimal values. Using Fekete's subadditive lemma we show that the N-th square roots of h(N) and s(N) have a limit as N goes to infinity and that these two limits coincide. We also derive tight bounds for the common value of the limits.
