Multi-base Representations of Integers: Asymptotic Enumeration and Central Limit Theorems
نویسنده
چکیده
In a multi-base representation of an integer (in contrast to, for example, the binary or decimal representation) the base (or radix) is replaced by products of powers of single bases. The resulting numeral system has desirable properties for fast arithmetic. It is usually redundant, which means that each integer can have multiple different digit expansions, so the natural question for the number of representations arises. In this paper, we provide a general asymptotic formula for the number of such multi-base representations of a positive integer n. Moreover, we prove central limit theorems for the sum of digits, the Hamming weight (number of non-zero digits, which is a measure of efficiency) and the occurrences of a fixed digits in a random representation.
منابع مشابه
On the Number of Multi-base Representations of an Integer
In a multi-base representation of an integer (in contrast to, for example, the binary or decimal representation) the base (or radix) is replaced by products of powers of single bases. The resulting numeral system is usually redundant, which means that each integer can have multiple different digit expansions. We provide a general asymptotic formula for the number of such multi-base representati...
متن کاملCentral Limit Theorems for Generalized Pólya Urn Models
In this paper we obtain central limit theorems for generalized Pólya urn models with L ≥ 2 colors where one out ofK different replacements (actions) is applied randomly at each step. Each possible action constitutes a row of the replacement matrix, which can be nonsquare and random. The actions are chosen following a probability distribution given by an arbitrary function of the proportions of ...
متن کاملEEH: AGGH-like public key cryptosystem over the eisenstein integers using polynomial representations
GGH class of public-key cryptosystems relies on computational problems based on the closest vector problem (CVP) in lattices for their security. The subject of lattice based cryptography is very active and there have recently been new ideas that revolutionized the field. We present EEH, a GGH-Like public key cryptosystem based on the Eisenstein integers Z [ζ3] where ζ3 is a primitive...
متن کاملLimit Theorems for the Number of Summands in Integer Partitions
Central and local limit theorems are derived for the number of distinct summands in integer partitions, with or without repetitions, under a general scheme essentially due to Meinardus. The local limit theorems are of the form of Cramér-type large deviations and are proved by Mellin transform and the two-dimensional saddle-point method. Applications of these results include partitions into posi...
متن کاملKernel Quantile Regression for Nonlinear Stochastic Models
We consider kernel quantile estimates for drift and scale functions in nonlinear stochastic regression models. Under a general dependence setting, we establish asymptotic point-wise and uniform Bahadur representations for the kernel quantile estimates. Based on those asymptotic representations, central limit theorems are obtained. Applications to nonlinear autoregressive models and linear proce...
متن کامل