The concepts of a number normal to base $s$ (simply normal) and a normal number (absolutely normal, i.e., normal to any natural base) were introduced by '{E}.~Borel (1909). These concepts are based on the notion of the frequency of a digit in the $s$-adic representation of a number. Although Borel proved that almost all numbers in the interval $[0, 1]$ (in the sense of Lebesgue measure) are normal, he did not construct a single example of a normal number. Later, this was done by A.~Lebesgue, W.~Sierpi'{n}ski, and others. These studies initiated a new direction in the metric theory of real numbers, which is known today as the theory of normal properties of numbers in various encoding (representation) systems.

In general, the theory of normal numbers in the $s$-adic numeral system has been developed in the works of '{E}.~ Borel, H.~Weyl, A.~Lebesgue, W.~Sierpi'{n}ski, D.~ Champernowne, S.~Pillai, D.~Wall, P.~Erd\H{o}s, A.~Copeland, H.~Davenport, and many others.

The $Q_s$-representation and $Q_s$-expansion of numbers from the interval $[0;1]$, introduced by M. V. Pratsiovytyi in 1986 as a generalization of the classical $s$-adic representation, have been extensively studied from topological, metric, and fractal viewpoints. Generalizations of the theorems of Borel, Besicovitch–Eggleston, Billingsley, and others were obtained within this framework. The $Q_s$-representation has found numerous applications in metric and probabilistic number theory, fractal analysis, the theory of singular distributions of random variables, and the study of functions with complicated local structure and fractal properties. In particular, it has been used by many authors to generalize constructions of singular continuous functions, nowhere monotone functions, and nowhere differentiable continuous functions.

The present paper continues the investigation of metric properties of the $Q_s$-representation of real numbers and the normality properties determined by their $Q_s$-representations. The main purpose of the paper is to apply the $Q_s$-representation to obtain a generalization of K. Nagasaki’s theorem on the superfractal nature of the set of weakly normal numbers that are not normal in the sense of Borel.

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