There are \( 2^p\) subsets of \( \mathbb F_p\) . 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 \( A+A \in \mathbb F_p\) , 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.
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