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Hotspot lemmas for noncompact spaces. (English) Zbl 1458.37003

The “hotspot lemma” is a tool which allows to establish the normality of a number in a digital numeration system. The classical setting is the \(b\)-ary expansion for a fixed \(b \geq 2\), as treated by I. I. Piatetski-Shapiro [Izv. Akad. Nauk SSSR, Ser. Mat. 15, 47–52 (1951; Zbl 0042.04902)]. In the present paper the authors establish a variant of the lemma which works in numeration systems with an unbounded number of digits, under the additional assumption of a tightness condition. This solves a problem in the statement of some theorems in [N. G. Moshchevitin and I. D. Shkredov, Math. Notes 73, No. 4, 539–550 (2003; Zbl 1173.37302); translation from Mat. Zametki 73, No. 4, 577–589 (2003)], where the problems that can be caused by an unbounded number of digits were not correctly taken into account.

MSC:

37A30 Ergodic theorems, spectral theory, Markov operators
37A44 Relations between ergodic theory and number theory
11J70 Continued fractions and generalizations
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References:

[1] Piatetskii-Shapiro, I. I., Izv. Akad. Nauk SSSR Ser. Mat., 15, 1, 47 (1951) · Zbl 0042.04902
[2] Postnikov, A. G., Trudy Mat. Inst. Steklov, 82, 0 (1966)
[3] Moshchevitin, N. G.; Shkredov, I. D., Math. Notes, 73, 3-4, 539 (2003) · Zbl 1173.37302 · doi:10.1023/A:1023263305857
[4] Postnikov, A. G., Trudy Mat. Inst. Steklov, 57, 0 (1960) · Zbl 0106.12101
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