Acknowledgements
This book would not have been possible without significant assistance
of many people. Any list we write down is guaranteed to be incom-
plete, but we will give it a try. First, the authors wish to thank Nets
Katz for contributing much of the material in Chapters 7 and 8. He
also explained to us the importance of this material within the con-
text of the Erd˝ os distance problem and its relatives. We also wish to
thank Misha Rudnev, whose collaboration with the second listed au-
thor on the finite field variant of the Erd˝ os-Falconer distance problem
ultimately led to the last three chapters of the book.
Numerous people have contributed important remarks on vari-
ous aspects of the book. We are particularly indebted to Bill Banks,
Pete Casazza, Jeremy Chapman, David Covert, Lacy Hardcastle, Der-
rick Hart, Tyler Salisbury-Jones, Doowon Koh, Mihalis Mourgoglou,
Laura Poe, Shannon Reed, Krystal Taylor, Ignacio Uriarte-Tuero, Lee
Anh Vinh, and Chandra Vaidyanthan.
The authors of the book were profoundly influenced in writing
of this book by their conversations with many brilliant mathemati-
cians who contributed to the study of the Erd˝ os distance conjec-
ture and related problems in the past 20 years. We have not had
the honor of interacting with nearly all of them, but we did learn
much from discussions with Michael Christ, Steve Hofmann, Philippe
Jaming, Nets Katz, Mihalis Kolountzakis, Sergei Konyagin, Izabella
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