Fourier restriction and well-approximable numbers

We use a deterministic construction to prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem for dimension $d=1$ and parameter range $0<a,b\le d$ and $b\le 2a$. Previous constructions by Hambrook and Łaba \cite{HL2013} and Chen \cite{chen} required randomness and only covered the range $0<b\le a\le d=1$. We also resolve a question of Seeger \cite{seeger-private} about the Fourier restriction inequality on the sets of well-approximable numbers.