A $ℤ^{d}$ generalization of the Davenport-Erdős construction of normal numbers

Mordechay Levin; Meir Smorodinsky

Displaying similar documents to “A $ℤ^{d}$ generalization of the Davenport-Erdős construction of normal numbers”

Growth of the product $\prod_{j = 1}^{n} (1 - x^{a_{j}})$

J. P. Bell, P. B. Borwein, L. B. Richmond (1998)

Acta Arithmetica

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We estimate the maximum of $\prod_{j = 1}^{n} | 1 - x^{a_{j}} |$ on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when $a_{j}$ is $j^{k}$ or when $a_{j}$ is a quadratic in j that takes on positive integer values, the maximum grows as exp(cn), where c is a positive constant. This complements results of Sudler and Wright that show exponential growth when $a_{j}$ is j. In contrast we show, under fairly general conditions, that the maximum is less than $2^{n} / n^{r}$ , where r is an arbitrary positive number. One consequence...

Ergodic averages and free $ℤ^{2}$ actions

Zoltán Buczolich (1999)

Fundamenta Mathematicae

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If the ergodic transformations S, T generate a free $ℤ^{2}$ action on a finite non-atomic measure space (X,S,µ) then for any $c_{1}, c_{2} \in ℝ$ there exists a measurable function f on X for which ${(N + 1)}^{- 1} \sum_{j = 0}^{N} f (S^{j} x) \to c_{1}$ and ${(N + 1)}^{- 1} \sum_{j = 0}^{N} f (T^{j} x) \to c_{2} µ$ -almost everywhere as N → ∞. In the special case when S, T are rationally independent rotations of the circle this result answers a question of M. Laczkovich.

Normal numbers and subsets of N with given densities

Haseo Ki, Tom Linton (1994)

Fundamenta Mathematicae

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For X ⊆ [0,1], let $D_{X}$ denote the collection of subsets of ℕ whose densities lie in X. Given the exact location of X in the Borel or difference hierarchy, we exhibit the exact location of $D_{X}$ . For α ≥ 3, X is properly $D_{ξ} (Π_{α}^{0})$ iff $D_{X}$ is properly $D_{ξ} (Π_{1 + α}^{0})$ . We also show that for every nonempty set X ⊆[0,1], $D_{X}$ is $Π_{3}^{0}$ -hard. For each nonempty $Π_{2}^{0}$ set X ⊆ [0,1], in particular for X = x, $D_{X}$ is $Π_{3}^{0}$ -complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to base n is $Π_{3}^{0}$ -complete. Moreover,...

On a discrete version of the antipodal theorem

Krzysztof Oleszkiewicz (1996)

Fundamenta Mathematicae

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The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping $f : S^{k} \to ℝ^{k}$ there exists a point $x \in S^{k}$ such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which $S^{k}$ is replaced by the set of vertices of a high-dimensional cube equipped with Hamming’s metric. In place of equality we obtain some optimal estimates of $i n f_{x} | | f (x) - f (- x) | |$ which were previously known (as far as the author knows) only for f linear (cf. [1]).

A note on strange nonchaotic attractors

Gerhard Keller (1996)

Fundamenta Mathematicae

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For a class of quasiperiodically forced time-discrete dynamical systems of two variables (θ,x) ∈ $T^{1} \times ℝ_{+}$ with nonpositive Lyapunov exponents we prove the existence of an attractor Γ̅ with the following properties: 1. Γ̅ is the closure of the graph of a function x = ϕ(θ). It attracts Lebesgue-a.e. starting point in $T^{1} \times ℝ_{+}$ . The set θ:ϕ(θ) ≠ 0 is meager but has full 1-dimensional Lebesgue measure. 2. The omega-limit of Lebesgue-a.e point in $T^{1} \times ℝ_{+}$ is $Γ ̅$ , but for a residual set of points in $T^{1} \times ℝ_{+}$ the omega...

Length of continued fractions in principal quadratic fields

Guillaume Grisel (1998)

Acta Arithmetica

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Let d ≥ 2 be a square-free integer and for all n ≥ 0, let $l ({(\sqrt d)}^{2 n + 1})$ be the length of the continued fraction expansion of ${(\sqrt d)}^{2 n + 1}$ . If ℚ(√d) is a principal quadratic field, then under a condition on the fundamental unit of ℤ[√d] we prove that there exist constants C₁ and C₂ such that $C ₁ {(\sqrt d)}^{2 n + 1} \geq l ({(\sqrt d)}^{2 n + 1}) \geq C ₂ {(\sqrt d)}^{2 n + 1}$ for all large n. This is a generalization of a theorem of S. Chowla and S. S. Pillai [2] and an improvement in a particular case of a theorem of [6].

Partition properties of subsets of Pκλ

Masahiro Shioya (1999)

Fundamenta Mathematicae

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Let κ > ω be a regular cardinal and λ > κ a cardinal. The following partition property is shown to be consistent relative to a supercompact cardinal: For any $f : \cup_{n < ω} {[X]}_{\subset}^{n} \to γ$ with $X \subset P_{κ} λ$ unbounded and 1 < γ < κ there is an unbounded Y ∪ X with $| f^{''} {[Y]}_{\subset}^{n} | = 1$ for any n < ω.

An extension of a theorem of Marcinkiewicz and Zygmund on differentiability

S. Mukhopadhyay, S. Mitra (1996)

Fundamenta Mathematicae

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Let f be a measurable function such that $Δ_{k} (x, h; f) = {O (| h |}^{λ})$ at each point x of a set E, where k is a positive integer, λ > 0 and $Δ_{k} (x, h; f)$ is the symmetric difference of f at x of order k. Marcinkiewicz and Zygmund [5] proved that if λ = k and if E is measurable then the Peano derivative $f_{(k)}$ exists a.e. on E. Here we prove that if λ > k-1 then the Peano derivative $f_{([λ])}$ exists a.e. on E and that the result is false if λ = k-1; it is further proved that if λ is any positive integer and if the approximate Peano...

Distribution of lattice points on hyperbolic surfaces

Vsevolod F. Lev (1996)

Acta Arithmetica

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Let two lattices $Λ^{'}, Λ^{''} \subset ℝ^{s}$ have the same number of points on each hyperbolic surface $| x ₁ . . . x_{s} | = C$ . We investigate the case when Λ’, Λ” are sublattices of $ℤ^{s}$ of the same prime index and show that then Λ’ and Λ” must coincide up to renumbering the coordinate axes and changing their directions.

Strongly meager sets and subsets of the plane

Janusz Pawlikowski (1998)

Fundamenta Mathematicae

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Let $X \subseteq 2^{w}$ . Consider the class of all Borel $F \subseteq X \times 2^{w}$ with null vertical sections $F_{x}$ , x ∈ X. We show that if for all such F and all null Z ⊆ X, $\cup_{x \in Z} F_{x}$ is null, then for all such F, $\cup_{x \in X} F_{x} \neq 2^{w}$ . The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].

Sumsets of Sidon sets

Imre Z. Ruzsa (1996)

Acta Arithmetica

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1. Introduction. A Sidon set is a set A of integers with the property that all the sums a+b, a,b∈ A, a≤b are distinct. A Sidon set A⊂ [1,N] can have as many as (1+o(1))√N elements, hence N/2 sums. The distribution of these sums is far from arbitrary. Erdős, Sárközy and T. Sós [1,2] established several properties of these sumsets. Among other things, in [2] they prove that A + A cannot contain an interval longer than C√N, and give an example that $N^{1 / 3}$ is possible. In [1] they show that...

The 2-Sylow subgroups of the tame kernel of imaginary quadratic fields

Hourong Qin (1995)

Acta Arithmetica

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1. Introduction. Let F be a number field and $O_{F}$ the ring of its integers. Many results are known about the group $K ₂ O_{F}$ , the tame kernel of F. In particular, many authors have investigated the 2-Sylow subgroup of $K ₂ O_{F}$ . As compared with real quadratic fields, the 2-Sylow subgroups of $K ₂ O_{F}$ for imaginary quadratic fields F are more difficult to deal with. The objective of this paper is to prove a few theorems on the structure of the 2-Sylow subgroups of $K ₂ O_{F}$ for imaginary quadratic fields F. In our Ph.D....

Shift spaces and attractors in noninvertible horseshoes

H. Bothe (1997)

Fundamenta Mathematicae

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As is well known, a horseshoe map, i.e. a special injective reimbedding of the unit square $I^{2}$ in $ℝ^{2}$ (or more generally, of the cube $I^{m}$ in $ℝ^{m}$ ) as considered first by S. Smale [5], defines a shift dynamics on the maximal invariant subset of $I^{2}$ (or $I^{m}$ ). It is shown that this remains true almost surely for noninjective maps provided the contraction rate of the mapping in the stable direction is sufficiently strong, and bounds for this rate are given.

The exceptional set of Goldbach numbers (II)

Hongze Li (2000)

Acta Arithmetica

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1. Introduction. A positive number which is a sum of two odd primes is called a Goldbach number. Let E(x) denote the number of even numbers not exceeding x which cannot be written as a sum of two odd primes. Then the Goldbach conjecture is equivalent to proving that E(x) = 2 for every x ≥ 4. E(x) is usually called the exceptional set of Goldbach numbers. In [8] H. L. Montgomery and R. C. Vaughan proved that $E (x) = O (x^{1 - Δ})$ for some positive constant Δ > 0 $. I n [3] C h e n a n d P a n p r o v e d t h a t o n e c a n t a k e Δ > 0.01 . I n [6], w e p r o v e d t h a t E (x) = O (x^{0.921})$ . In this paper we prove the following...

Polynomial cycles in certain local domains

T. Pezda (1994)

Acta Arithmetica

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1. Let R be a domain and f ∈ R[X] a polynomial. A k-tuple $x ₀, x ₁, . . ., x_{k - 1}$ of distinct elements of R is called a cycle of f if $f (x_{i}) = x_{i + 1}$ for i=0,1,...,k-2 and $f (x_{k - 1}) = x ₀$ . The number k is called the length of the cycle. A tuple is a cycle in R if it is a cycle for some f ∈ R[X]. It has been shown in [1] that if R is the ring of all algebraic integers in a finite extension K of the rationals, then the possible lengths of cycles of R-polynomials are bounded by the number $7^{7 \cdot 2^{N}}$ , depending only on the degree N of K. In this...

A complement to the theory of equivariant finiteness obstructions

Paweł Andrzejewski (1996)

Fundamenta Mathematicae

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It is known ([1], [2]) that a construction of equivariant finiteness obstructions leads to a family $w_{α}^{H} (X)$ of elements of the groups $K_{0} (ℤ [π_{0} {(W H (X))}_{α}^{*}])$ . We prove that every family $w_{α}^{H}$ of elements of the groups $K_{0} (ℤ [π_{0} {(W H (X))}_{α}^{*}])$ can be realized as the family of equivariant finiteness obstructions $w_{α}^{H} (X)$ of an appropriate finitely dominated G-complex X. As an application of this result we show the natural equivalence of the geometric construction of equivariant finiteness obstruction ([5], [6]) and equivariant generalization of Wall’s...

A combinatorial approach to partitions with parts in the gaps

Dennis Eichhorn (1998)

Acta Arithmetica

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Many links exist between ordinary partitions and partitions with parts in the “gaps”. In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let $p_{k, m} (j, n)$ be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let $p *_{k, m} (j, n)$ be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then $p *_{k, m} (j, n) = p_{k, m} (j, n)$ .

The Lucas congruence for Stirling numbers of the second kind

Roberto Sánchez-Peregrino (2000)

Acta Arithmetica

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0. Introduction. The numbers introduced by Stirling in 1730 in his Methodus differentialis [11], subsequently called “Stirling numbers” of the first and second kind, are of the greatest utility in the calculus of finite differences, in number theory, in the summation of series, in the theory of algorithms, in the calculation of the Bernstein polynomials [9]. In this study, we demonstrate some properties of Stirling numbers of the second kind similar to those satisfied by binomial coefficients;...

Displaying similar documents to “A ℤ d generalization of the Davenport-Erdős construction of normal numbers”

Displaying similar documents to “A $ℤ^{d}$ generalization of the Davenport-Erdős construction of normal numbers”