A basis of ℤₘ, II

Min Tang; Yong-Gao Chen

Displaying similar documents to “A basis of ℤₘ, II”

On a conjecture of Dekking : The sum of digits of even numbers

Iurie Boreico, Daniel El-Baz, Thomas Stoll (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $q \geq 2$ and denote by $s_{q}$ the sum-of-digits function in base $q$ . For $j = 0, 1, \dots, q - 1$ consider $# {0 \leq n < N : s_{q} (2 n) \equiv j (mod q)} .$ In 1983, F. M. Dekking conjectured that this quantity is greater than $N / q$ and, respectively, less than $N / q$ for infinitely many $N$ , thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.

A basis of Zₘ

Min Tang, Yong-Gao Chen (2006)

Colloquium Mathematicae

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Let $σ_{A} (n) = | (a, a^{'}) \in A ² : a + a^{'} = n |$ , where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, $σ_{A} (n)$ is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which $σ_{A} (n)$ is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and $σ_{A} (n ̅) \leq 768$ for all n̅ ∈ Zₘ.

Addendum to "Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces" (Colloq. Math. 127 (2012), 105-109)

Aydin Sh. Shukurov (2014)

Colloquium Mathematicae

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It is well known that if φ(t) ≡ t, then the system ${φ ⁿ (t)}_{n = 0}^{\infty}$ is not a Schauder basis in L₂[0,1]. It is natural to ask whether there is a function φ for which the power system ${φ ⁿ (t)}_{n = 0}^{\infty}$ is a basis in some Lebesgue space $L_{p}$ . The aim of this short note is to show that the answer to this question is negative.

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Claire Voisin (2013)

Annales scientifiques de l'École Normale Supérieure

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We prove that Bloch’s conjecture is true for surfaces with $p_{g} = 0$ obtained as $0$ -sets $X_{σ}$ of a section $σ$ of a very ample vector bundle on a variety $X$ with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as $0$ on holomorphic $2$ -forms of $X_{σ}$ , then it acts as $0$ on $0$ -cycles of degree $0$ of $X_{σ}$ . In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general $X_{σ}$ ...

Invariance of the parity conjecture for $p$ -Selmer groups of elliptic curves in a $D_{2 p^{n}}$ -extension

Thomas de La Rochefoucauld (2011)

Bulletin de la Société Mathématique de France

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We show a $p$ -parity result in a $D_{2 p^{n}}$ -extension of number fields $L / K$ ( $p \geq 5$ ) for the twist $1 \oplus η \oplus τ$ : $W (E / K, 1 \oplus η \oplus τ) = {(- 1)}^{〈1 \oplus η \oplus τ, X_{p} (E / L)〉}$ , where $E$ is an elliptic curve over $K$ , $η$ and $τ$ are respectively the quadratic character and an irreductible representation of degree $2$ of $Gal (L / K) = D_{2 p^{n}}$ , and $X_{p} (E / L)$ is the $p$ -Selmer group. The main novelty is that we use a congruence result between $ε_{0}$ -factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the $p$ -parity conjecture...

Characterization of the alternating groups by their order and one conjugacy class length

Alireza Khalili Asboei, Reza Mohammadyari (2016)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group, and let $N (G)$ be the set of conjugacy class sizes of $G$ . By Thompson’s conjecture, if $L$ is a finite non-abelian simple group, $G$ is a finite group with a trivial center, and $N (G) = N (L)$ , then $L$ and $G$ are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation)....

On coarse embeddability into $ℓ_{p}$ -spaces and a conjecture of Dranishnikov

Piotr W. Nowak (2006)

Fundamenta Mathematicae

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We show that the Hilbert space is coarsely embeddable into any $ℓ_{p}$ for 1 ≤ p ≤ ∞. It follows that coarse embeddability into ℓ₂ and into $ℓ_{p}$ are equivalent for 1 ≤ p < 2.

A variation of Thompson's conjecture for the symmetric groups

Mahdi Abedei, Ali Iranmanesh, Farrokh Shirjian (2020)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group and let $N (G)$ denote the set of conjugacy class sizes of $G$ . Thompson’s conjecture states that if $G$ is a centerless group and $S$ is a non-abelian simple group satisfying $N (G) = N (S)$ , then $G ≅ S$ . In this paper, we investigate a variation of this conjecture for some symmetric groups under a weaker assumption. In particular, it is shown that $G ≅ Sym (p + 1)$ if and only if $| G | = (p + 1)!$ and $G$ has a special conjugacy class of size $(p + 1)! / p$ , where $p > 5$ is a prime number. Consequently, if $G$ is a centerless group with $N (G) = N (Sym (p + 1))$ , then...

Results related to Huppert’s $ρ$ - $σ$ conjecture

Xia Xu, Yong Yang (2023)

Czechoslovak Mathematical Journal

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We improve a few results related to Huppert’s $ρ$ - $σ$ conjecture. We also generalize a result about the covering number of character degrees to arbitrary finite groups.

Thompson’s conjecture for the alternating group of degree $2 p$ and $2 p + 1$

Azam Babai, Ali Mahmoudifar (2017)

Czechoslovak Mathematical Journal

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For a finite group $G$ denote by $N (G)$ the set of conjugacy class sizes of $G$ . In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N (G) = N (L)$ , then $G ≅ L$ . We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z (G) = 1$ and $N (G) = N (A_{i})$ is necessarily isomorphic to $A_{i}$ , where $i \in {2 p, 2 p + 1}$ .

On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

Ke-Pao Lin, Xue Luo, Stephen S.-T. Yau, Huaiqing Zuo (2014)

Journal of the European Mathematical Society

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It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function $ψ (x, y)$ which is the number of positive integers $\leq x$ and free of prime factors $> y$ . Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional ( $n \geq 3$ ) real right-angled simplices. In this...

Three-space problems and bounded approximation properties

Wolfgang Lusky (2003)

Studia Mathematica

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Let ${R ₙ}_{n = 1}^{\infty}$ be a commuting approximating sequence of the Banach space X leaving the closed subspace A ⊂ X invariant. Then we prove three-space results of the following kind: If the operators Rₙ induce basis projections on X/A, and X or A is an $ℒ_{p}$ -space, then both X and A have bases. We apply these results to show that the spaces $C_{Λ} = \bar{s p a n} z^{k} : k \in Λ \subset C ()$ and $L_{Λ} = \bar{s p a n} z^{k} : k \in Λ \subset L ₁ ()$ have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.

Normal number constructions for Cantor series with slowly growing bases

Dylan Airey, Bill Mance, Joseph Vandehey (2016)

Czechoslovak Mathematical Journal

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Let $Q = {(q_{n})}_{n = 1}^{\infty}$ be a sequence of bases with $q_{i} \geq 2$ . In the case when the $q_{i}$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$ -Cantor series expansion is both $Q$ -normal and $Q$ -distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$ , and from this construction we can provide computable constructions of numbers with atypical normality properties. ...

On the domination of triangulated discs

Noor A&amp;#039;lawiah Abd Aziz, Nader Jafari Rad, Hailiza Kamarulhaili (2023)

Mathematica Bohemica

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Let $G$ be a $3$ -connected triangulated disc of order $n$ with the boundary cycle $C$ of the outer face of $G$ . Tokunaga (2013) conjectured that $G$ has a dominating set of cardinality at most $\frac{1}{4} (n + 2)$ . This conjecture is proved in Tokunaga (2020) for $G - C$ being a tree. In this paper we prove the above conjecture for $G - C$ being a unicyclic graph. We also deduce some bounds for the double domination number, total domination number and double total domination number in triangulated discs.

$L_{p, q}$ spaces

Joseph Kupka

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CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p, q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p, q} (B)$ ........ 235. Examples in $L_{p, q}$ theory...................................................................................

Asymptotic values of modular multiplicities for ${GL}_{2}$

Sandra Rozensztajn (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the irreducible constituents of the reduction modulo $p$ of irreducible algebraic representations $V$ of the group ${Res}_{K / ℚ_{p}} {GL}_{2}$ for $K$ a finite extension of $ℚ_{p}$ . We show that asymptotically, the multiplicity of each constituent depends only on the dimension of $V$ and the central character of its reduction modulo $p$ . As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.

Manin’s and Peyre’s conjectures on rational points and adelic mixing

Alex Gorodnik, François Maucourant, Hee Oh (2008)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$ . We prove Manin’s conjecture on the asymptotic (as $T \to \infty$ ) of the number of $K$ -rational points of $X$ of height less than $T$ , and give an explicit construction of a measure on $X (𝔸)$ , generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $𝐆 (K)$ on $X (𝔸)$ . Our approach is based on the mixing property of $L^{2} (𝐆 (K) ∖ 𝐆 (𝔸))$ which we obtain with a rate of convergence. ...

A note on representation functions with different weights

Zhenhua Qu (2016)

Colloquium Mathematicae

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For any positive integer k and any set A of nonnegative integers, let $r_{1, k} (A, n)$ denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ and $r_{1, l} (A, n) = r_{1, l} (ℕ ∖ A, n)$ hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ for all n ≥ n₀, we have $r_{1, k} (A, n) \to \infty$ as n → ∞.

Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces

Aydin Sh. Shukurov (2012)

Colloquium Mathematicae

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A necessary condition for Kostyuchenko type systems and system of powers to be a basis in $L_{p}$ (1 ≤ p < +∞) spaces is obtained. In particular, we find a necessary condition for a Kostyuchenko system to be a basis in $L_{p}$ (1 ≤ p < +∞).

On the $r$ -free values of the polynomial $x^{2} + y^{2} + z^{2} + k$

Gongrui Chen, Wenxiao Wang (2023)

Czechoslovak Mathematical Journal

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Let $k$ be a fixed integer. We study the asymptotic formula of $R (H, r, k)$ , which is the number of positive integer solutions $1 \leq x, y, z \leq H$ such that the polynomial $x^{2} + y^{2} + z^{2} + k$ is $r$ -free. We obtained the asymptotic formula of $R (H, r, k)$ for all $r \geq 2$ . Our result is new even in the case $r = 2$ . We proved that $R (H, 2, k) = c_{k} H^{3} + O (H^{9 / 4 + ε})$ , where $c_{k} > 0$ is a constant depending on $k$ . This improves upon the error term $O (H^{7 / 3 + ε})$ obtained by G.-L. Zhou, Y. Ding (2022).