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Digital expansion of exponential sequences. (English) Zbl 1072.11006

Let \(a\) and \(q\) be integers larger than one such that \(\log_a q\) is irrational. The author studies the average number of occurences of a (\(s+1\))-digit subblock in the \(q\)-ary digital expansions of \(a^n\), \(1\leq n\leq N\). The first result concerns the most significant digits: It is shown that there exists a positive real constant \(\gamma\) such that the average number of occurrences of the subblock in the most significant \(\gamma\log_q N\) digits equals \(\gamma\log_q N q^{-(s+1)}+O(1)\).
For the second result concerning the least significant digits, \(q\) is assumed to be a prime not dividing \(a\). Under this stronger assumption, the bounds on the error term have a lower order of magnitude. Let \(A_1(N)\), \(A_2(N)\) be positive integer-valued functions with \[ (\log_q N)^\eta\leq A_1(N)\leq A_2(N)\leq (\log_q N)^{3/2-\varepsilon} \] for some positive constants \(\varepsilon\) and \(\eta\). Then the average number of occurrences of the given subblock among the digits with indices between \(A_1(N)\) and \(A_2(N)\) equals the “right” main term \((A_2(N)-A_1(N))q^{-(s+1)}\) plus an error term of \(O(1/\log^\lambda N)\) for some positive \(\lambda\).
The proofs use a discrepancy bound due to P. Kiss and R. F. Tichy [Proc. Japan Acad., Ser. A 65, 135–138 (1989; Zbl 0692.10041); ibid. 65, 191–194 (1989; Zbl 0692.10041)] and bounds for exponential sums.
In both cases, lower bounds for the number of subblock changes and the sum-of-digits function immediately follow from the results.

MSC:

11A63 Radix representation; digital problems
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11K38 Irregularities of distribution, discrepancy
11L07 Estimates on exponential sums

Citations:

Zbl 0692.10041
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References:

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