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The Distribution of Generalized Sum-of-Digits Functions in Residue Classes

✍ Scribed by Abigail Hoit

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
184 KB
Volume
79
Category
Article
ISSN
0022-314X

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✦ Synopsis

Let Q=[Q j ] j=0 be a strictly increasing sequence of integers with Q 0 =1 and such that each Q j is a divisor of Q j+1 . The sequence Q is a numeration system in the sense that every positive integer n has a unique base-Q'' representation of the form n= j 0 a j (n) Q j with digits'' a j (n) satisfying 0

where n= j 0 a j (n) Q j is the base-Q representation of n and the component functions f j are defined on [0, 1, ..., Q j+1 Γ‚Q j &1] and satisfy f j (0)=0. We study the distribution of integer-valued Q-additive functions in residue classes. Our main result gives necessary and sufficient conditions for f to be uniformly (resp. non-uniformly) distributed modulo m, for any given prime m. We apply this result to many cases, showing, for example, that the sum-of-digits functions associated with base-Q representations are uniformly distributed modulo any prime m.

In this paper we generalize this problem in three directions. The first generalization is to consider more general ``numeration systems'' than the binary system. These systems are defined as follows. Let Q=[Q j ] j=0 be a

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