scholarly journals On the distribution of the convergents of almost all real numbers

1970 ◽  
Vol 2 (4) ◽  
pp. 425-441 ◽  
Author(s):  
P Erdös
Keyword(s):  
Real Numbers ◽  
Almost All

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 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.

scholarly journals Reconstructing Subsets of Reals

10.37236/1452 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
A. J. Radcliffe ◽  
A. D. Scott
Keyword(s):  
Finite Subset ◽  
Fixed Size ◽  
Real Numbers ◽  
Locally Finite ◽  
Finite Set ◽  
Almost All

We consider the problem of reconstructing a set of real numbers up to translation from the multiset of its subsets of fixed size, given up to translation. This is impossible in general: for instance almost all subsets of $\mathbb{Z}$ contain infinitely many translates of every finite subset of $\mathbb{Z}$. We therefore restrict our attention to subsets of $\mathbb{R}$ which are locally finite; those which contain only finitely many translates of any given finite set of size at least 2. We prove that every locally finite subset of $\mathbb{R}$ is reconstructible from the multiset of its 3-subsets, given up to translation.

scholarly journals On the Discrepancy of Quasi-Progressions

10.37236/828 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Sujith Vijay
Keyword(s):  
Lower Bound ◽  
Real Number ◽  
Positive Density ◽  
Real Numbers ◽  
Almost All ◽  
Beatty Sequence

A quasi-progression, also known as a Beatty sequence, consists of successive multiples of a real number, with each multiple rounded down to the largest integer not exceeding it. In 1986, Beck showed that given any $2$-colouring, the hypergraph of quasi-progressions contained in $\{0,1,\ldots,n \}$ corresponding to almost all real numbers in $(1, \infty)$ have discrepancy at least $\log^{*} n$, the inverse of the tower function. We improve the lower bound to $(\log n)^{1/4 - o(1)}$, and also show that there is some quasi-progression with discrepancy at least $(1/50) n^{1/6}$. The results remain valid even if the $2$-colouring is replaced by a partial colouring of positive density.

Over de Taal Van de Wiskunde

1984 ◽  
Vol 19 ◽  
pp. 31-39
Author(s):  
R.P. Nederpelt
Keyword(s):  
Natural Language ◽  
Artificial Language ◽  
Mathematical Language ◽  
Linguistic Features ◽  
Real Numbers ◽  
Mathematical Text ◽  
Full Development ◽  
The One ◽  
Almost All ◽  
Special Language

The professional mathematician employs in his writings a special language, which we call 'the language of mathematics'. It has two components: on the one hand (a fragment of) natural language, and on the other hand a highly specialized artificial language. The latter part attracts the eye in a mathematical text, because of its deviating form: it contains symbols, formulae and the like. Yet almost all peculiarities of mathematical language, including those of the artificial part, can be embedded in the usual natural language frame. This different appearance of mathematical language originates from a mathematicians urge for efficiency, clarity and compactness. A noteworthy advantage of mathematical language is found in its treatment of coreferences. The artificial part of mathematical language employs a highly developed reference mechanism, called binding. A formula like 'εx IR(x > x2)' for instance, contains a bound variable 'x' of which only the category (viz. the set of real numbers) is fixed. The variable 'x' can be replaced by another one without changing the meaning of the formula: 'yε IR(y > y2)'.This mechanism of binding makes all kinds of referential words, as used in natural language (such as relative pronouns), superfluous. Mathematical language is still in full development. Especially at the word and word group level, many interesting linguistic features can be observed. At the sentence and text level, however, mathematical language is not very revolutionary. Here improvements are possible, and some of these are proposed in the article.

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