Failure of Sobolev surjectivity for a screen with accumulating gaps

The energy-space invertibility of a screen single-layer operator need not extend to the full range of shifted Sobolev trace spaces. We construct a compact Ahlfors regular one-dimensional screen consisting of intervals that accumulate geometrically at the origin. For this screen there is an exponent \(\alpha\in(0,1/2)\) such that, for every positive Helmholtz wavenumber, the single-layer operator fails to be surjective from \(\hH^{t-1/2}\) to \(\hH^{t+1/2}\) whenever \(\alpha<t<1/2\). The example disproves the all-\(d\)-set assertion in Conjecture 4.8 of Caetano, Chandler-Wilde, Gibbs, Hewett and Moiola, through its permitted case \(n=d=1\). A positive harmonic function on the complement of the repeating screen has a radial exponent below the usual square-root edge exponent. Its finite boundary charge belongs to the energy space but violates the claimed higher Sobolev regularity, although its Helmholtz single-layer image has an \(H^1\) extension. The noninteger-dimensional case is not addressed.