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 \(d=1\) $d=1$ and parameter range \(0 < a,b \le d\) $0<a,b\le d$ and \(b\le 2a\) $b\le 2a$ . Previous constructions by Hambrook and Łaba [15] and Chen [5] required randomness and only covered the range \(0 < b \le a \le d=1\) $0<b\le a\le d=1$ . We also resolve a question of Seeger [29] about the Fourier restriction inequality on the sets of well-approximable numbers.