Generalized Beatty sequences

A. McD. Mercer

International Journal of Mathematics and Mathematical Sciences, 1978, vol. 1, 1-4

Abstract:

A well-known result due to S . Beatty is that if α and β are positive irrational numbers satisfying α − 1 + β − 1 = 1 then each positive integer is to be found in precisely one of the sequences { [ k α ] } , { [ k β ] } ( k = 1 , 2 , 3 , … ) where [ x ] denotes the integral part of x . The present note generalizes this result to the case of the pair of sequences { [ f ( k ) ] } , { [ g ( k ) ] } with suitable hypotheses on the functions f and g . The special case f ( x ) = α x , g ( x ) = β x is the result due to Beatty.

Date: 1978
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:284925

DOI: 10.1155/S0161171278000514

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