Hotspot lemmas for noncompact spaces (Q2210389)

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 \textit{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 [\textit{N. G. Moshchevitin} and \textit{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.