Space-filling curves and small-ball probabilities for the Kloosterman process

We construct explicitly a space-filling curve in the support of the limiting Kloosterman process, using the planar partial-sum curve of the Rudin–Shapiro sequence. The qualitative existence statement was previously attributed to J. Bober by Kowalski. The resulting curve is a closed, $1\mathrm{/}2$-Hölder continuous loop whose image contains a specified triangle of area $1\mathrm{/}128$. Its Fourier coefficients are purely imaginary and satisfy $\mathrm{\mid }\widehat{f}(h)\mathrm{\mid }\le (2\pi \mathrm{\mid }h\mathrm{\mid }{)}^{-1}$ for $h\mathrm{\ne }0$. We also prove a uniform lower bound for small balls centred at closed $1\mathrm{/}2$-Hölder curves with a fixed coefficient bound, of the form $\exp \{-C{\epsilon }^{-2}{\log }^2(2\mathrm{/}\epsilon )\}$. Known functional convergence then gives a quantitative lower bound for the limiting proportion of two-parameter Kloosterman paths approximating the constructed loop.