On Pisier Type Theorems

Abstract

AbstractFor any integer $$h\geqslant 2$$ h ⩾ 2 , a set of integers $$B=\{b_i\}_{i\in I}$$ B = { b i } i ∈ I is a $$B_h$$ B h -set if all h-sums $$b_{i_1}+\ldots +b_{i_h}$$ b i 1 + … + b i h with $$i_1<\ldots <i_h$$ i 1 < … < i h are distinct. Answering a question of Alon and Erdős [2], for every $$h\geqslant 2$$ h ⩾ 2 we construct a set of integers X which is not a union of finitely many $$B_h$$ B h -sets, yet any finite subset $$Y\subseteq X$$ Y ⊆ X contains an $$B_h$$ B h -set Z with $$|Z|\geqslant \varepsilon |Y|$$ | Z | ⩾ ε | Y | , where $$\varepsilon :=\varepsilon (h)$$ ε : = ε ( h ) . We also discuss questions related to a problem of Pisier about the existence of a set A with similar properties when replacing $$B_h$$ B h -sets by the requirement that all finite sums $$\sum _{j\in J}b_j$$ ∑ j ∈ J b j are distinct.