Hausdorff dimension of the graph of the Weierstrass function

For parameters $0 < a < 1$ and $b > 1$ with $ab > 1$, the classical Weierstrass function is

$W_{a,b}(x) = \sum_{n=0}^{\infty} a^{n} \cos(2\pi b^{n} x).$

It is continuous on $\mathbb{R}$ and, by Hardy's 1916 sharpening of Weierstrass's 1872 example, nowhere differentiable whenever $0 < a < 1$ and $ab \ge 1$. Its graph is a compact subset of $\mathbb{R}^2$, and the claim at issue is that its Hausdorff dimension is

$\dim_{H} \operatorname{graph}(W_{a,b}) = 2 + \log_{b} a = 2 + \frac{\log a}{\log b},$

a number strictly between $1$ and $2$, since $0 < a < 1$ gives $\log_b a < 0$ and $ab > 1$ gives $\log_b a > -1$.

Status: settled in 2018 for integer $b$, and open in general. Shen proved the formula for every integer $b \ge 2$ and every $a \in (1/b, 1)$. For non-integer $b$ the value of the Hausdorff dimension of the graph remains open. Even in the settled range the result is far out of reach here: it needs Hausdorff measure and dimension, hyperbolic dynamics, and the absolute continuity of an SRB measure on a solenoidal attractor.

Not proved in this library. Nothing here rests on the value of this dimension, and no page may cite the formula as established.

What is known, and what would settle the rest. The box-counting dimension of the graph is $2 + \log_b a$ and has been classical for decades (see Falconer, The Geometry of Fractal Sets); since Hausdorff dimension never exceeds box dimension, the entire difficulty is the lower bound. The successive advances were: Hunt (1998), who proved the formula for the randomly phased variant $\sum a^n \cos(2\pi(b^n x + \theta_n))$ for almost every phase sequence $(\theta_n)$; Barański, Bárány and Romanowska (2014), who proved it for integer $b$ and $a$ above a threshold $a_b \in (1/b, 1)$; and Shen (2018), who removed the threshold and covered every integer $b \ge 2$ and every $a \in (1/b, 1)$, by proving that the SRB measure of the associated solenoidal attractor is absolutely continuous. Ren and Shen (2021) then generalised the formula from $\cos$ to an arbitrary real analytic periodic $\varphi$, as a dichotomy: for integer $b \ge 2$ and $a \in (1/b, 1)$, either $\sum a^n \varphi(b^n x)$ is real analytic or its graph has Hausdorff dimension $2 + \log_b a$. Every one of these results keeps the hypothesis that $b$ is an integer. What would settle what remains is the non-integer case, where the associated dynamics is no longer a self-affine expanding map of a circle by an integer degree and the current method does not apply.

Why it matters here. Continuity and nowhere differentiability of a Weierstrass function are in scope for this library and will be proved from the Weierstrass M-test and a direct oscillation estimate. The dimension of the graph is not, and the gap between the two is worth naming: the elementary statement that the curve has no tangent anywhere is a nineteenth-century theorem, while the quantitative statement of how rough it is took until 2018 and is still incomplete. Recording that here keeps the nowhere-differentiability page from implying more than it proves.