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 ≤ d and $$b\le 2a$$ b ≤ 2 a . 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 ≤ a ≤ d = 1 . We also resolve a question of Seeger [29] about the Fourier restriction inequality on the sets of well-approximable numbers.