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There are 2^p subsets of F_p. All of them are sumsets,
that is sets of the form A+B:={a+b: a in A, b in B}, since
A=A+{0}. However, Green and Ruzsa showed that there are only
2^(p/3+o(p)) sumsets A+A in F_p, and Noga Alon,
Adrian Ubis and I show that that there are no more than
1.97^p sumsets A+B, with |A|,|B|>1. Moreover we show that there
are only 2^(p/2+o(p)) sumsets A+B with |A|,|B| > m(p),
if m(p) goes to infinity with p (though arbitrarily slowly).
One can ask more precise questions; for example, how many sumsets
A+B are there with |A|=k and |B|=l ? We get bounds on this, in
particular showing that, when k,l > m(p) (as above),
one gets 2^(p/2+o(p)) such sumsets if and only if k+l = p/4 + o(p).
The methods used are combinatorial and analytic.
Different approaches to the distribution of primes
The Princeton Companion to Mathematics: Analytic number theory
Prime number patterns (2009 Ford Prize)
It is easy to determine whether a given integer is prime (2008 Chauvenet Prize)
Prime number races (2007 Ford Prize)
It's as easy as abc
Zaphod Beeblebrox's brain and the fifty-ninth row of Pascal's triangle (1995 Hasse Prize)
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