A Remark on Sieving in Biased Coin Convolutions

Periodica Mathematica Hungarica

In this note we construct an algorithm generating any discrete distribution with an arbitrary coin (and, as a result, with arbitrary initial distribution). The coin need not be fair and the target distribution can be supported on a countable set.

Arxiv preprint cs/0401019, 2004

Abstract: While it is well known that a Turing machine equipped with the ability to flip a fair coin cannot compute more that a standard Turing machine, we show that this is not true for a biased coin. Indeed, any oracle set $ X $ may be coded as a probability $ p_ {X} $ such that if a Turing machine is given a coin which lands heads with probability $ p_ {X} $ it can compute any function recursive in $ X $ with arbitrarily high probability. We also show how the assumption of a non-recursive bias can be weakened by using a sequence of ...

Glasgow Mathematical Journal, 2004

Let µ p be the distribution of a random variable on the interval [0, 1), each digit of whose binary expansion is 0 or 1 with probability p or 1 − p. Thus µ p = * ∞ n=1 (pδ 0 + (1 − p)δ 1 2 n). We show that for any Borel subsets E, F of [0, 1) we have λ(E + F) ≥ µ p (E) α µ q (F) β , where 0 < α, β < 1 with α log a + β log b = log 2 and a = [max{p, 1 − p}] −1 , b = [max{q, 1 − q}] −1. Here λ = µ 1/2 denotes Lebesgue measure.

Acta Arithmetica, 1995

Information Processing Letters, 1995

1980

The large volume of material forces us to restrict ourselves to certain directions only. The choice of theorems and bibliographic references is entirely subjective and does not pretend to completeness. The basic problem of probabilistic number theory is the study of the distribution of values of arithmetic functions, i.e., functions defined on the set of natural numbers N. The classic investigation, taking its start in the works of K. F. Gauss and P. G. Lejeune-Dirichlet, from the point of view of the modern theory did not exceed the study of mathematical expectation in sequences of probability spaces {En, Un, ~n}, as n § Here En = [l,...,n}, U n is the set of all subsets of E n and n 1 ~ 1 A~U.. v. (A)= g m=l m~A Most arithmetic functions (a.f.) considered in number theory have the property of additivity or multiplicativity. An a.f. h(m) (respectively, g(m)) is called additive (multiplicative) if for any pair of relatively prime numbers m and n, h(m n)=h(m)+h(n) (g(m n) =g(m)g(n)). Let p be a prime. If h(p ~) =h(p) and g(pa) =g(p) for a>-l, then h(m) and g(m) are called strongly additive (s.a.) and strongly multiplicative (s.m.) a.f., respectively. Throughout the entire text following we restrict ourselves to them when this simplifies the formulation. Transition to the general case does not involve major difficulties. Starting in 1917 there have appeared papers devoted to the study of the asymptotic behavior of the statistical distribution functions ~,,(x)=v. (m<<.n, h(m)-~(n) <X) (n) where h(m) is a real a.a.f, and ~(n) and ~(n) are certain normalizing sequences. This problem occupied a central place in the problems of today. The history of the question is given in the monographs [i, 2] and in the surveys [3-5]. In papers up to the sixth decade, elementary and combinatorial methods dominated, allowing the study of separate examples of a.f. or their rather narrow classes. Only the Erdos-Wintner theorem gave an answer about the conditions on convergence of ~n(X) to the limiting function of the distribution for ~(n)= 0 and B(n)-i for any real a.a.f, h(m). 2. Kubilyus' Method In 1954-1956 I. Kubilyus developed a method allowing one to reduce the study of vn(x) to classical limit theorems of probability theory [i]. We shall explain its essence briefly. Stressing the generality of the method, we carry out the argument for a sequence of functions. This extends the class of limit laws (cf. also [7]). We represent the sequence of real s.a.a.f, hn(m) in the form h. (m) = ~] h~ (m),

Journal of the London Mathematical Society, 1983

Bolyai Society Mathematical Studies, 2013

LMS J. Comput. Math, 2010

Let β ∈ (1, 2) be a Pisot number and let H β denote Garsia's entropy for the Bernoulli convolution associated with β. Garsia, in 1963 showed that H β < 1 for any Pisot β. For the Pisot numbers which satisfy x m = x m−1 + x m−2 + · · · + x + 1 (with m ≥ 2) Garsia's entropy has been evaluated with high precision by Alexander and Zagier for m = 2 and later by Grabner Kirschenhofer and Tichy for m ≥ 3, and it proves to be close to 1. No other numerical values for H β are known.

1997

Combinatorial search methods often exhibit a large variability in performance. We study the cost prooles of combinatorial search procedures. Our study reveals some intriguing properties of such cost prooles. The distributions are often characterized by very long tails or \heavy tails&quot;. We will show that these distributions are best characterized by a general class of distributions that have no moments (i.e., an innnite mean, variance , etc.). Such non-standard distributions have recently been observed in areas as diverse as economics, statistical physics, and geophysics. They are closely related to fractal phenomena, whose study was introduced by Mandelbrot. We believe this is the rst nding of these distributions in a purely computational setting. We also show how random restarts can effectively eliminate heavy-tailed behavior, thereby dramatically improving the overall performance of a search procedure. This paper has not already been accepted by and is not currently under rev...