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# Averaging along Uniform Random Integers

Author(s): Élise Janvresse | Thierry de la Rue

Journal: Uniform Distribution Theory
ISSN 1336-913X

Volume: 7;
Issue: 2;
Start page: 35;
Date: 2012;
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Keywords: Benford's law | log-density | \$H_infty\$-density | uniform random integers

ABSTRACT
Motivated by giving a meaning to “The probability that a random integer has initial digit \$d\$”, we define a emph{URI-set} as a random set \$E\$ of natural integers such that each \$≥1\$ belongs to \$E\$ with probability \$1/n\$, independently of other integers. This enables us to introduce two notions of densities on natural numbers: The emph{URI-density}, obtained by averaging along the elements of \$E\$, and the emph{local URI-density}, which we get by considering the \$k\$-th element of \$E\$ and letting \$k\$ go to \$infty\$. We prove that the elements of \$E\$ satisfy Benford's law, both in the sense of URI-density and in the sense of local URI-density. Moreover, if \$b_1\$ and \$b_2\$ are two multiplicatively independent integers, then the mantissae of a natural number in base \$b_1\$ and in base \$b_2\$ are independent. Connections of URI-density and local URI-density with other well-known notions of densities are established: Both are stronger than the natural density, and URI-density is equivalent to \$log\$-density. We also give a stochastic interpretation, in terms of URI-set, of the \$H_infty\$-density.