Substitutions and 1/2-discrepancy of {n θ+ x}

الأصل · EN

The sequence of 1/2-discrepancy sums of {x + i θ 1} is realized through a sequence of substitutions on an alphabet of three symbols; particular attention is paid to x=0. The first application is to show that any asymptotic growth rate of the discrepancy sums not trivially forbidden may be achieved. A second application is to show that for badly approximable θ and any x the range of values taken over i=0,1,...n-1 is asymptotically similar to (n), a stronger conclusion than given by the Denjoy-Koksma inequality.