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

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

2011-05-29

arxiv.org/abs/1105.5810v1

Mathematics

Dynamical Systems

In review

Scientific paper

The sequence of 1/2-discrepancy sums of $\{x + i \theta \bmod 1\}$ 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 $\theta$ and any $x$ the range of values taken over $i=0,1,...n-1$ is asymptotically similar to $\log(n)$, a stronger conclusion than given by the Denjoy-Koksma inequality.

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