*This beautiful book is about how to estimate large
quantities—and why. Building on nothing more than first-year calculus, it
goes all the way into deep asymptotical methods and shows how these can
be used to solve problems in number theory, combinatorics, probability, and
geometry. The author is a master of exposition: starting from such a simple
fact as the infinity of primes, he leads the reader through small steps, each
carefully motivated, to many theorems that were cutting-edge when discovered,
and teaches the general methods to be learned from these results.*

—**László Lovász,
Eötvös-Loránd University**

*This is a lovely little travel guide to a country you
might not even have heard about - full of wonders, mysteries, small and large
discoveries ... and in Joel Spencer you have the perfect travel guide!*

—**Günter M. Ziegler, Freie Universität
Berlin, coauthor of "Proofs from THE BOOK"**

Asymptotics in one form or another are part of
the landscape for every mathematician. The objective of this book is
to present the ideas of how to approach asymptotic problems that arise
in discrete mathematics, analysis of algorithms, and number theory. A
broad range of topics is covered, including distribution of prime
integers, Erdős Magic, random graphs, Ramsey numbers, and
asymptotic geometry.

The author is a disciple of Paul Erdős, who taught him about
Asymptopia. Primes less than $n$, graphs with $v$
vertices, random walks of $t$ steps—Erdős was
fascinated by the limiting behavior as the variables approached, but
never reached, infinity. Asymptotics is very much an art. The various
functions $n\ln n$, $n^2$, $\frac{\ln n}{n}$,
$\sqrt{\ln n}$, $\frac{1}{n\ln n}$ all have distinct
personalities. Erdős knew these functions as personal
friends. It is the author's hope that these insights may be passed on,
that the reader may similarly feel which function has the right
temperament for a given task. This book is aimed at strong
undergraduates, though it is also suitable for particularly good high
school students or for graduates wanting to learn some basic
techniques.

Asymptopia is a beautiful world. Enjoy!

Readership

Undergraduate and graduate students interested in asymptotic techniques.