On Pisier Type Theorems

Abstract

For any integer \(h\geqslant 2\), a set of integers \(B=\{b_i\}_{i\in I}\) is a \(B_h\)-set if all h-sums \(b_{i_1}+\ldots +b_{i_h}\) with \(i_1<\ldots <i_h\) are distinct. Answering a question of Alon and Erdős [2], for every \(h\geqslant 2\) we construct a set of integers X which is not a union of finitely many \(B_h\)-sets, yet any finite subset \(Y\subseteq X\) contains an \(B_h\)-set Z with \(|Z|\geqslant \varepsilon |Y|\), where \(\varepsilon :=\varepsilon (h)\). We also discuss questions related to a problem of Pisier about the existence of a set A with similar properties when replacing \(B_h\)-sets by the requirement that all finite sums \(\sum _{j\in J}b_j\) are distinct.