Introduction

This survey is a written and slightly expanded version of the talk given at the ICMS workshop on motivic integration and its interactions with model theory and non-archimedean geometry. Its presence may seem to require a few preliminary words of explanation: not only does the title apparently fail to reflect any of the three components of that of the conference itself, but also the author is far from being an expert in any of these. However, one must remember that there is but a small step from summation to integration. Moreover, as I will argue in the last section, there are some basic problems in the theory of exponential sums (and their applications) for which it seems not impossible that logical ideas could be useful, and hence presenting the context to model-theorists in particular could well be useful. In fact, the most direct connection between exponential sums and the topics of the workshop will be a survey of the extension to exponential sums of the beautiful counting results of Chatzidakis, van den Dries and Macintyre's [CDM]. These may also have some further applications.

Acknowledgements. I wish to thank warmly the organizers of the workshop for preparing a particularly rich and inspiring program, and for inviting me to participate.