scholarly journals Problems in Strong Uniform Distribution

2014 ◽  
Vol 59 (1) ◽  
pp. 51-64
Author(s):  
Kwo Chan ◽  
Radhakrishnan Nair
Keyword(s):  
Real Number ◽  
Uniform Distribution ◽  
Lebesgue Measure ◽  
Fractional Part ◽  
Positive Lebesgue Measure ◽  
The Real ◽  
Almost All

Abstract In 1923 A. Khinchin asked if given any B ⊆ [0, 1) of positive Lebesgue measure, we have #{n : 1 ≤ n ≤ N : {nx} ∈ B} → |B| for almost all x with respect to Lebesgue measure. Here {y} denotes the fractional part of the real number y and |A| denotes the Lebesgue measure of the set A in [0, 1). In 1970 J. Marstrand showed the answer is no. In this paper the authors survey contributions to this subject since then.

scholarly journals The discrepancy of some real sequences

2003 ◽  
Vol 93 (2) ◽  
pp. 268
Author(s):  
H. Kamarul Haili ◽  
R. Nair
Keyword(s):  
Real Number ◽  
Lebesgue Measure ◽  
Fractional Part ◽  
Real Numbers ◽  
The Real ◽  
Almost Everywhere

Let $(\lambda_n)_{n\geq 0}$ be a sequence of real numbers such that there exists $\delta > 0$ such that $|\lambda_{n+1} - \lambda_n| \geq \delta , n = 0,1,...$. For a real number $y$ let $\{ y \}$ denote its fractional part. Also, for the real number $x$ let $D(N,x)$ denote the discrepancy of the numbers $\{ \lambda _0 x \}, \cdots , \{ \lambda _{N-1} x \}$. We show that given $\varepsilon > 0$, 9774 D(N,x) = o ( N^{-\frac{1}{2}}(\log N)^{\frac{3}{2} + \varepsilon})9774 almost everywhere with respect to Lebesgue measure.

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.

Axioms of symmetry: Throwing darts at the real number line

1986 ◽  
Vol 51 (1) ◽  
pp. 190-200 ◽  
Author(s):  
Chris Freiling
Keyword(s):  
Real Number ◽  
Lebesgue Measure ◽  
Thought Experiment ◽  
Continuum Hypothesis ◽  
Number Line ◽  
Formal Proof ◽  
The Real ◽  
Joint Measurability ◽  
The Axiom Of Choice ◽  
Real Number Line

AbstractWe will give a simple philosophical “proof” of the negation of Cantor's continuum hypothesis (CH). (A formal proof for or against CH from the axioms of ZFC is impossible; see Cohen [1].) We will assume the axioms of ZFC together with intuitively clear axioms which are based on some intuition of Stuart Davidson and an old theorem of Sierpiński and are justified by the symmetry in a thought experiment throwing darts at the real number line. We will in fact show why there must be an infinity of cardinalities between the integers and the reals. We will also show why Martin's Axiom must be false, and we will prove the extension of Fubini's Theorem for Lebesgue measure where joint measurability is not assumed. Following the philosophy—if you reject CH you are only two steps away from rejecting the axiom of choice (AC)—we will point out along the way some extensions of our intuition which contradict AC.

scholarly journals A metrical result on the discrepancy of (nα)

1998 ◽  
Vol 40 (3) ◽  
pp. 393-425 ◽  
Author(s):  
J. Schoissengeier
Keyword(s):  
Characteristic Function ◽  
Real Number ◽  
Lebesgue Measure ◽  
Fractional Part ◽  
Irrational Numbers ◽  
Metrical Result ◽  
Image Position

In the following let Ω be the set of irrational numbers in the interval [0,1] and let λ be Lebesgue measure restricted to Ω. For any real number x, let {x} = x - [x] be the fractional part of x. Let N be anatural number and let α e Ω. Thenis known as the discrepancy of the sequence (nα)n>1 modulo 1; here c[x, y) denotes the characteristic function of the interval [x, y).

scholarly journals β-transformation, invariant measure and uniform distribution

1999 ◽  
Vol 66 (3) ◽  
pp. 418-428
Author(s):  
Gavin Brown ◽  
Qinghe Yin
Keyword(s):  
Invariant Measure ◽  
Uniform Distribution ◽  
Lebesgue Measure ◽  
Almost All

AbstractLet Tβ be the β-transformation on [0, 1). When β is an integer Tβ is ergodic with respect to Lebesgue measure and almost all orbits {} are uniformly distributed. Here we consider the non-integer case, determine when Tα, Tβ have the same invariant measure and when (appropriately normalised) orbits are uniformly distributed.

Waterborne Outbreaks in Sweden – Causes and Etiology

1986 ◽  
Vol 18 (10) ◽  
pp. 185-190 ◽  
Author(s):  
Y. Andersson ◽  
T. A. Stenström
Keyword(s):  
Real Number ◽  
Unknown Etiology ◽  
Raw Water ◽  
Actual Number ◽  
Single Family ◽  
The Real ◽  
Almost All ◽  
Community Outbreaks ◽  
Microbial Agents

Thirty-two waterborne outbreaks in Sweden are known during 1975 - 1984, affecting nearly 12 000 people. These range from single family outbreaks to community outbreaks affecting up to 3 000 people. Microbial agents have been isolated in about 40 % of the outbreaks, the rest are of unknown etiology. Epidemiological investigations have shown that only a fraction of the actual number of cases are initially reported. The real number as judged from epidemiological follow-up investigations was in many instances tenfold higher. In almost all the cases, the cause of the outbreaks are technical deficiencies like back-siphonage of wastewater along drainage pipes, broken sewerage or sudden pollution of raw water intakes coinciding with malfunction of chlorination.

Sequences with bounded logarithmic discrepancy

1990 ◽  
Vol 107 (2) ◽  
pp. 213-225 ◽  
Author(s):  
R. C. Baker ◽  
G. Harman
Keyword(s):  
Real Number ◽  
Fractional Part ◽  
Real Sequence ◽  
The Real ◽  
Image Position

Let {α} denote the fractional part of the real number α. Write χ(x, y) = 1 for {x} < y and χ(x, y) = 0 otherwise. A real sequence (xn) is uniformly distributed (mod 1) ifIt is a consequence of (1·1) that DN = o(N),Whereis the discrepancy of the sequence (xn).

scholarly journals Exceptional Sets in Uniform Distribution

1979 ◽  
Vol 22 (2) ◽  
pp. 145-160 ◽  
Author(s):  
R. C. Baker
Keyword(s):  
Uniform Distribution ◽  
Lebesgue Measure ◽  
Fractional Part ◽  
Real Numbers ◽  
Exceptional Sets ◽  
Measurable Set ◽  
Image Position

Let B be a measurable set of real numbers in (0,1) of Lebesgue measure |B| and let x1, …, xn be real. Thendenotes the number of j (1 ≦j≦n) for which the fractional part {xj}∈B. The discrepancy of x1, …, xn iswhere the supremum is taken over all intervals I in [0,1].

Waterborne Outbreaks in Sweden – Causes and Etiology

1987 ◽  
Vol 19 (3-4) ◽  
pp. 575-580 ◽  
Author(s):  
Y. Andersson ◽  
T. A. Stenström
Keyword(s):  
Real Number ◽  
Unknown Etiology ◽  
Raw Water ◽  
Actual Number ◽  
Single Family ◽  
The Real ◽  
Almost All ◽  
Community Outbreaks ◽  
Microbial Agents

Thirty-two waterborne outbreaks in Sweden are known during 1975 - 1984, affecting nearly 12000 people. These range from single family outbreaks to community outbreaks affecting up to 3000 people. Microbial agents have been isolated in about 40 % of the outbreaks, the rest are of unknown etiology. Epidemiological investigations have shown that only a fraction of the actual number of cases are initially reported. The real number as judged from epidemiological follow-up investigations was in many instances tenfold higher. In almost all the cases, the cause of the outbreaks are technical deficiencies like back-siphonage of wastewater along drainage pipes, broken sewerage or sudden pollution of raw water intakes coinciding with malfunction of chlorination.

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