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Benford's Law for the 3 x + 1 Function
Author(s) -
Lagarias Jeffrey C.,
Soundararajan K.
Publication year - 2006
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610706023131
Subject(s) - iterated function , benford's law , sequence (biology) , mathematics , combinatorics , function (biology) , base (topology) , assertion , discrete mathematics , mathematical analysis , statistics , computer science , biology , genetics , evolutionary biology , programming language
Benford's law (to base B ) for an infinite sequence { x k : k ⩾ 1} of positive quantities x k is the assertion that {log B x k : k ⩾ 1} is uniformly distributed (mod 1). The 3 x + 1 function T ( n ) is given by T ( n ) = (3 n + 1)/2 if n is odd, and T ( n ) = n /2 if n is even. This paper studies the initial iterates x k = T ( k ) ( x 0 ) for 1 ⩽ k ⩽ N of the 3 x + 1 function, where N is fixed. It shows that for most initial values x 0 , such sequences approximately satisfy Benford's law, in the sense that the discrepancy of the finite sequence {log B x k : 1 ⩽ k ⩽ N } is small.