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Friday, September 18, 2009 |
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Sum-Product Problem: New Generalisations and Applications |
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The celebrated sum-product theorem of Bourgain-Katz-Tao asserts that for any set
$A$ in a prime finite field (such that the cardinality $|A|$ is not too small or
large) at least one of the sets $A + A = \{a_1 + a_2 : a_1, a_2 \in A \}$ and $AA
= \{a_1 a_2 : a_1, a_2 \in A \}$ is of cardinality at least $|A|^{1+ \epsilon}$
for some fixed $\epsilon > 0$. Various explicit versions (with an explicit value
of $\epsilon$) of this result have been obtained recently, and the result has also
been extended in various directions including different functions on sets, such as
$A^{-1} + A^{-1} = \{a_1 ^{-1} + a_2^{-1} : a_1, a_2 \in A \}$. We outline some
new applications of such results to various number-theoretic problems, including
estimating the ``concentration'' of rational points on some curves over prime
finite fields $F_p$ and the number of solutions of congruences of the type $x^x =
1 \bmod{p}$ for a prime $p$. We also describe some recent generalisations to the
settings of groups of points on elliptic curves and of matrix rings. |