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In this paper it is shown that if $E\subset\mathbb R^{n+1}$ is an $s$-ADregular compact set, with $s\in [n-\frac12,n)$, and $E$ is contained in ahyperplane or, more generally, in an $n$-dimensional $C^1$ manifold, then theHausdorff dimension of the harmonic measure for the domain $\mathbbR^{n+1}\setminus E$ is strictly smaller than $s$, i.e., than the Hausdorffdimension of $E$.

The dimension of harmonic measure on some AD-regular flat sets of fractional dimension