Self-affine functions stretch differently in different directions — horizontal and vertical scales are decoupled. Built from the Q_s-representation of real numbers (a generalization of base expansions where digit probabilities vary), these functions inherit the number-theoretic complexity of their construction.
The paper on Hölder exponents and fractal structure of self-affine functions (arXiv: 2603.24411) computes local regularity at each point in terms of the asymptotic digit frequencies in its Q_s-expansion, and shows that the level sets — the preimages of single values — can be fractal.
The Hölder exponent at a point measures how rough the function is locally: a large exponent means smooth, a small one means jagged. For these self-affine functions, the exponent at each point is determined by which digits appear in the point's Q_s-representation and how often. Points with atypical digit frequencies have atypical regularity. The set of points achieving maximum value can itself be a fractal set — not a single point or an interval, but a Cantor-like dust.
The through-claim: the regularity of a function is distributed by the arithmetic of its domain. The Hölder exponent at a point is not a local geometric property — it is determined by the global number-theoretic structure of the point's representation. The function's smoothness at x is a statement about the digits of x. Arithmetic governs analysis.
2603.24411. Analysis / self-affine functions / Hölder exponents / fractal geometry / Q_s-representation.