Efficient Distributed Coin-tossing Protocols

We extend a theorem of L. E. Dubins on “purely finitely additive disintegrations” of measures (cf. [4]) and apply this result to the disintegrations of extremal Gibbs states with respect to the asymptotic algebra enlarging another result of L. E. Dubins on the symmetric coin tossing game.We recall the following definition of L. E. Dubins (cf. [3], [4]): Let (X , , μ) be a measure space, a sub σ-algebra of . A real function σx (A), is called a measurable-disintegration of μ if:(i) ∀x ∊ X , σx(.) is a finitely additive measure .(ii) ∀A ∊ , σ. (A) is constant on each -atom.(iii) For each A ∊ , σ. (A) is measurable with respect to the completion of by μ and (iv)σx(B) = 1 if x ∊ B ∊ .

Download Full-text

On the Fourier transform of coin-tossing type measures

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2019.123706 ◽

2020 ◽

Vol 484 (1) ◽

pp. 123706 ◽

Cited By ~ 1

Author(s):

Xiang Gao ◽

Jihua Ma ◽

Kunkun Song ◽

Yanfang Zhang

Keyword(s):

Fourier Transform ◽

Coin Tossing ◽

The Fourier Transform

Download Full-text

Topological Horseshoes and Coin-Tossing Dynamics

Chaotic Dynamics in Nonlinear Theory ◽

10.1007/978-81-322-2092-3_2 ◽

2014 ◽

pp. 29-53

Author(s):

Lakshmi Burra

Keyword(s):

Coin Tossing ◽

Topological Horseshoes

Download Full-text

NOTE ON FOURIER–STIELTJES COEFFICIENTS OF COIN-TOSSING MEASURES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972720000313 ◽

2020 ◽

Vol 102 (3) ◽

pp. 479-489

Author(s):

XIANG GAO ◽

SHENGYOU WEN

Keyword(s):

Number Theory ◽

Lower Bound ◽

Transcendental Number ◽

Linear Forms In Logarithms ◽

Multiplicative Semigroup ◽

Linear Forms ◽

Coin Tossing

It is known that the Fourier–Stieltjes coefficients of a nonatomic coin-tossing measure may not vanish at infinity. However, we show that they could vanish at infinity along some integer subsequences, including the sequence ${\{b^{n}\}}_{n\geq 1}$ where $b$ is multiplicatively independent of 2 and the sequence given by the multiplicative semigroup generated by 3 and 5. The proof is based on elementary combinatorics and lower-bound estimates for linear forms in logarithms from transcendental number theory.

Download Full-text