On Strong Normality

2016 ◽  
Vol 11 (1) ◽  
pp. 59-78 ◽  
Author(s):  
Jean-Marie De Koninck ◽  
Imre Kátai ◽  
Bui Minh Phong
Keyword(s):  
Normal Numbers ◽  
Real Numbers ◽  
Strong Normality ◽  
Almost All

AbstractWe introduce the concept of strong normality by defining strong normal numbers and provide various properties of these numbers, including the fact that almost all real numbers are strongly normal.

scholarly journals Higher Order Oscillation and Uniform Distribution

2019 ◽  
Vol 14 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Shigeki Akiyama ◽  
Yunping Jiang
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Uniform Distribution ◽  
Differentiable Function ◽  
Mild Condition ◽  
Higher Order ◽  
Real Numbers ◽  
Mobius Function ◽  
Real Polynomials ◽  
Almost All

AbstractIt is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form (e2πiαβn g(β))n∈𝕅, for a non-decreasing twice differentiable function g with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number α and almost all real numbers β> 1 (alternatively, for a fixed real number β> 1 and almost all real numbers α) and for all real polynomials Q(x), sequences (αβng(β)+ Q(n)) n∈𝕅 are uniformly distributed modulo 1.

On the typical size and cancellations among the coefficients of some modular forms

2018 ◽  
Vol 166 (1) ◽  
pp. 173-189
Author(s):  
FLORIAN LUCA ◽  
MAKSYM RADZIWIŁŁ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Binary Quadratic Form ◽  
Cusp Forms ◽  
Real Numbers ◽  
Standard Argument ◽  
Typical Size ◽  
Holomorphic Cusp Forms ◽  
Weak Hypothesis ◽  
Almost All

AbstractWe obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato–Tate density. Examples of such sequences come from coefficients of severalL-functions of elliptic curves and modular forms. In particular, we show that |τ(n)| ⩽n11/2(logn)−1/2+o(1)for a set ofnof asymptotic density 1, where τ(n) is the Ramanujan τ function while the standard argument yields log 2 instead of −1/2 in the power of the logarithm. Another consequence of our result is that in the number of representations ofnby a binary quadratic form one has slightly more than square-root cancellations for almost all integersn.In addition, we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato–Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally and might be within reach unconditionally using the currently established potential automorphy.

A note on the largest digits in Lüroth expansion

2014 ◽  
Vol 10 (04) ◽  
pp. 1015-1023 ◽  
Author(s):  
Luming Shen ◽  
Yiying Yu ◽  
Yuxin Zhou
Keyword(s):  
New York ◽  
Hausdorff Dimension ◽  
Infinite Series ◽  
Real Numbers ◽  
Lecture Notes ◽  
Lüroth Expansion ◽  
Almost All

It is well known that every x ∈ (0, 1] can be expanded into an infinite Lüroth series with the form of [Formula: see text] where dn(x) ≥ 2 and is called the nth digits of x for each n ≥ 1. In [Representations of Real Numbers by Infinite Series, Lecture Notes in Mathematics, Vol. 502 (Springer, New York, 1976)], Galambos showed that for Lebesgue almost all x ∈ (0, 1], [Formula: see text], where Ln(x) = max {d1(x), …, dn(x)} denotes the largest digit among the first n ones of x. In this paper, we consider the Hausdorff dimension of the set [Formula: see text] for any α ≥ 0.

scholarly journals NORMAL NUMBERS AND COMPLETENESS RESULTS FOR DIFFERENCE SETS

2017 ◽  
Vol 82 (1) ◽  
pp. 247-257 ◽  
Author(s):  
KONSTANTINOS A. BEROS
Keyword(s):  
Ergodic Theory ◽  
Difference Sets ◽  
Normal Numbers ◽  
Real Numbers ◽  
Image Position

AbstractWe consider some natural sets of real numbers arising in ergodic theory and show that they are, respectively, complete in the classes ${\cal D}_2 \left( {{\bf{\Pi }}_3^0 } \right)$ and ${\cal D}_\omega \left( {{\bf{\Pi }}_3^0 } \right)$, that is, the class of sets which are 2-differences (respectively, ω-differences) of ${\bf{\Pi }}_3^0 $ sets.

scholarly journals The Pelger-Huët Anomaly and Megaloblastic Anemia

Blood ◽  
1963 ◽  
Vol 22 (4) ◽  
pp. 472-476 ◽  
Author(s):  
S. ARDEMAN ◽  
I. CHANARIN ◽  
A. W. FRANKLAND
Keyword(s):  
Vitamin B12 ◽  
Pernicious Anemia ◽  
Peripheral Blood ◽  
Megaloblastic Anemia ◽  
Normal Numbers ◽  
Sex Chromatin ◽  
Almost All

Abstract A patient is described with addisonian pernicious anemia and with the Pelger-Huët anomaly of leukocytes. Before the patient was treated with vitamin B12, her peripheral blood contained three- and four-lobed neutrophils, but with therapy almost all the neutrophils showed the characteristic bilobed form of the Pelger-Huët anomaly. Before treatment the sex chromatin appendage was present in the neutrophils in normal numbers, but these could not be identified after treatment.

scholarly journals Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real?

Erkenntnis ◽  
2019 ◽  
Author(s):  
Nicolas Gisin
Keyword(s):  
Quantum Theory ◽  
Initial Conditions ◽  
Classical Mechanics ◽  
Bohmian Mechanics ◽  
Physical Reality ◽  
Real Numbers ◽  
Quantum Theories ◽  
Amount Of Information ◽  
Classical Chaos ◽  
Almost All

Abstract It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is empirically equivalent to classical mechanics, but uses only finite-information numbers. This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic. Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality.

scholarly journals VARIATIONS AROUND A PROBLEM OF MAHLER AND MENDÈS FRANCE

2012 ◽  
Vol 92 (1) ◽  
pp. 37-44
Author(s):  
YANN BUGEAUD
Keyword(s):  
General Question ◽  
Real Numbers ◽  
Almost All

AbstractWe discuss the following general question and some of its extensions. Let (εk)k≥1 be a sequence with values in {0,1}, which is not ultimately periodic. Define ξ:=∑ k≥1εk/2k and ξ′:=∑ k≥1εk/3k. Let 𝒫 be a property valid for almost all real numbers. Is it true that at least one among ξ and ξ′ satisfies 𝒫?

scholarly journals Higher-rank Bohr sets and multiplicative diophantine approximation

2019 ◽  
Vol 155 (11) ◽  
pp. 2214-2233 ◽  
Author(s):  
Sam Chow ◽  
Niclas Technau
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Structural Theory ◽  
Real Numbers ◽  
Successive Minima ◽  
Arbitrary Rank ◽  
Littlewood Conjecture ◽  
Higher Rank ◽  
Almost All

Gallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.

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