Represent a number in base s using digits 0 through s-1, and every number in [0,1] has a unique expansion (almost everywhere). Now allow one extra digit — use digits 0 through s, giving s+1 symbols for a base-s system. The expansion becomes redundant: most numbers can be written in multiple ways. The extra digit introduces degeneracy.
Pratsiovytyi, Karvatskyi, and Makarchuk (arXiv:2603.11310, 2026) study the probability distributions that arise when each digit in such a redundant expansion is drawn independently with fixed probabilities. The resulting random variable is an infinite Bernoulli convolution — a sum of scaled independent random variables. The central question: when is the distribution absolutely continuous (smooth, with a density function) and when is it singular (concentrated on a fractal set of measure zero)?
The answer depends on the digit probabilities. For some choices, the redundancy fills in gaps and the distribution is smooth. For others, the redundancy creates a Cantorval — a support set that is neither a full interval nor a classical Cantor set, but a hybrid: a union of intervals and Cantor-like components. The phase transition between smooth and singular distributions is controlled by how the redundancy is distributed across digits.
The structural point: redundancy in representation does not simply add flexibility. It creates a bifurcation in the character of the resulting object. With exactly the right number of symbols, the distribution is determined. With too many symbols, the distribution's nature becomes parameter-dependent — it can be either continuous or fractal depending on how the extra degree of freedom is used. The spare digit is not neutral. It forces a choice between smoothness and singularity that did not exist without it.