The relationship between Helicobacter pylori infection and chronic dyspepsia is controversial. To determine the effect of H. pylori infection on dyspeptic symptoms, we compared the prevalence of H. pylori infection in immigrants from developing countries and people born in industrialized countries. Upper abdominal symptoms were assessed by a questionnaire and H. pylori infection was determined with a 13C-urea breath test and serology. H. pylori infection was found in 63% of subjects from developing countries and 11% of subjects from industrialized countries. There was no difference in the prevalence of dyspeptic symptoms between the 2 groups. The lack of difference in chronic dyspeptic symptoms between the groups, despite a major difference in the H. pylori prevalence, suggests that H. pylori infection is not a major contributor to chronic dyspepsia.

We obtain a Fourier dimension estimate for sets of exact approximation order introduced by Bugeaud for certain approximation functions $\psi $. This Fourier dimension estimate implies that these sets of exact approximation order contain normal 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.

We show for decreasing, positive approximation functions $\psi $ such that $\tau = \lim _{q \to \infty } -\frac{\log \psi (q)}{\log q} < \frac{13 + \sqrt{73}}{8}$ and such that $q^{2} \psi (q) \to 0$ that the set $\operatorname{Exact}(\psi )$ of numbers approximable to the exact order $\psi $ has positive Fourier dimension. This implies that the set $\operatorname{Exact}(\psi )$ contains normal numbers.