The binary and ternary harmonic measures of the Cantor set are mutually singular

Let $E$ be the middle-thirds Cantor set and, for an integer $b\ge 2$, let $\nu_b$ be the measure on $E$ obtained by pushing forward the harmonic measure of simple random walk on the tree of $b$-adic intervals that meet $E$. The measure $\nu_3$ is the Cantor measure. Lyons and Peres asked whether $\nu_2$ and $\nu_3$ are mutually singular. We show that they are. The proof compares the conditional splits of the two measures on dyadic cells through Hellinger affinities. At one explicit dyadic cell, whose intersection with $E$ is a single ternary cylinder, the Cantor measure splits evenly while the harmonic measure, computed as an electrical current, does not; an exact computation with rational arithmetic bounds the harmonic split away from $1/2$, and the discrepancy persists for all nearby positions and scales of the cylinder. A Fourier argument resting on Stewart's theorem on digit changes in the bases $2$ and $3$ then shows that, if the two measures were not singular, such cells would carry a fixed share of their overlap at infinitely many levels, which is incompatible with the summability of the Hellinger losses.