Fourier restriction and well-approximable numbers

Abstract

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.

28 Pages. We fixed errors in the proofs of the following lemmas: Lemma 4.1: The estimate on the second-last sum has been fixed. This changed the statement of the lemma and one equation in the proof of Lemma 4.2. Lemma 3.8: Equation (3.21) has been fixed. Lemma 4.2: We introduce a constant $C_F$ in order to fix an error in the proof of equation (4.3). The main result is unchanged