Absolute continuity of the vertical distribution of the derivative

Prove that the pushforward measure of Lebesgue measure on (0,1/2) under the derivative D(y)=\partial_+\Phi(y) is absolutely continuous with respect to Lebesgue measure and has a square-integrable density; equivalently, establish the existence of a nonnegative function f\in L^1(\mathbb R)\cap L^2(\mathbb R) representing this measure.

Background

The paper studies the right derivative D(y)=\partial_+\Phi(y) of the zeta-star correspondence \Phi between binary coordinates and infinite-depth multiple zeta-star values. Although D is right-continuous everywhere and has a precisely characterized set of continuity points, it exhibits substantial arithmetic oscillation: it has jumps at dyadic points, infinite upper right Dini derivative at every interior point, and a fractal graph of dimension \log_2(8/3).

The authors define \rho=D_*m as the pushforward of Lebesgue measure m on (0,1/2) under D. The conjecture asks whether this vertical-value distribution is absolutely continuous and admits an L2 density, analogous to absolute-continuity phenomena for Bernoulli convolutions. The paper’s graph-dimension arguments use a Bernoulli convolution with contraction parameter 2/3, but explicitly do not establish absolute continuity or a uniform density estimate for the resulting vertical distribution.

References

Inspired by Solomyak's main results , we propose the following conjecture. The measure \rho is absolutely continuous with respect to Lebesgue measure and has a square-integrable density. More precisely, there exists a nonnegative function f\in L1()\cap L2() such that, for every Borel set A\subset, \begin{equation}\label{eq:global-vertical-density} \bigl({y\in(0,1/2):D(y)\in A}\bigr)=\int_A f(u)\,du. \end{equation} In particular, \int_{}f(u)\,du=\frac{1}{2}, \qquad \int_{}f(u)2\,du<+\infty.

eq:global-vertical-density:

({y∈(0,1/2):D(y)∈A})=∫Af(u) du.\bigl(\{y\in(0,1/2):D(y)\in A\}\bigr) =\int_A f(u)\,du.

— The derivative function of the zeta-star correspondence  (2609.31249 - Li, 25 Sep 2026) in Conjecture 1.1, Section 1, subsection “Main results”