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Digital expansion of exponential sequences

Michael Fuchs — 2002

Journal de théorie des nombres de Bordeaux

We consider the q -ary digital expansion of the first N terms of an exponential sequence a n . Using a result due to Kiss and Tichy [8], we prove that the average number of occurrences of an arbitrary digital block in the last c log N digits is asymptotically equal to the expected value. Under stronger assumptions we get a similar result for the first ( log N ) 3 2 - ϵ digits, where ϵ is a positive constant. In both methods, we use estimations of exponential sums and the concept of discrepancy of real sequences modulo 1 ...

On Kurzweil's 0-1 law in inhomogeneous Diophantine approximation

Michael Fuchs;  Dong Han Kim — 2016

Acta Arithmetica

We give a necessary and sufficient condition such that, for almost all s ∈ ℝ, ||nθ - s|| < ψ(n) for infinitely many n ∈ ℕ, where θ is fixed and ψ(n) is a positive, non-increasing sequence. This can be seen as a dual result to classical theorems of Khintchine and Szüsz which dealt with the situation where s is fixed and θ is random. Moreover, our result contains several earlier ones as special cases: two old theorems of Kurzweil, a theorem of Tseng and a recent...

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