J Austral Math Soc Ser A 51 pp216--236, 1991.

Digit Patterns and Transcendental Numbers

Patrick Morton and W. J. Mourant

Abstract

We use a theorem of Loxton and van der Poorten to prove the transcendence of certain real numbers defined by digit patterns. Among the results we prove are the following. If k is an integer at least 2, P is any nonzero pattern of digits base k, and e_P^(r)(n) Î [0, r - 1] counts the number of occurrences (mod r) of P in the base k representation of n, then m(e_P^(r)) = å^¥_n=0e_P^(r)(n)/rⁿ is transcendental except when k = 3, P = 1 and r = 2. When (r, k - 1) = 1 the linear span of the numbers m(e_P^(r)) has infinite dimension over Q, where P ranges over all patterns base k without leading zeros.

1980 AMS Subject Classification (1985 Revision): 10F35, 11A63

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Authors

Patrick Morton

Wellesley College, Wellesley, Massachusetts 02181, U.S.A.

W. J. Mourant

37 William J. Heights, Framingham, Massachusetts 01701, U.S.A.