When \(p>2\), let \(S\) be a set of integers and consider the inequality \(\|f\|_{L^p(\mathbb{T})} \leq C(p) \|f\|_{L^2(\mathbb{T})}\) where \(f(x) = \sum_{n \in S} a_n e^{2\pi i n x}\) and \(C(p)\) is a constant depends on \(p\). For various sets \(S\), it has been studied the range \(p\) where the inequality holds for any \(f\).
However, in the opposite direction, we can fix \(p\) and think of a set \(S\) which satisfies the inequality. This is called a \(\Lambda(p)\)-set. In this talk we discuss properties of \(\Lambda(p)\)-sets and the \(\Lambda(p)\)-set problem. We will also discuss variants of \(\Lambda(p)\)-sets which is defined in terms of Orlicz norms.