Preface

Summary

What the book is about

In 1976 I gave a new proof to the Grothendieck (two-dimensional) inequality. The proof, pushed a little further, yielded extensions of the inequality to higher dimensions. These extensions, in turn, revealed ‘Cartesian products in fractional dimensions’, and led in a setting of harmonic analysis to the solution of the (so-called) p-Sidon set problem. The solution subsequently gave rise to an index of combinatorial dimension, a general measurement of interdependence with connections to harmonic, functional, and stochastic analysis. In 1993 I was ready to tell the story, and began teaching topics courses about this work. The notes for these courses eventually became this book.

Broadly put, the book is about ‘dimensionality’. There are several interrelated themes, sub-themes, variations on themes. But at its very core, there is the notion that when we do mathematics – whatever mathematics we do – we start with independent building blocks, and build our constructs. Or, from an observer's viewpoint – not that of a builder – we assume existence of building blocks, and study structures we see. In either case, these are the questions: How are building blocks used, or put together? How complex are the constructs we build, or the structures we observe? How do we gauge, or detect, complexity? The answers involve notions of dimension.

The book is a mix of harmonic analysis, functional analysis, and probability theory.