Sum-Product Problem: New Generalisations and Applications
| Speaker: | Igor Shparlinski |
|---|---|
| Affiliation: | Macquarie University |
| Room: |
Mathematics & Computer Building (MC) 5158 |
Abstract:
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.
Finally, we mention several open problems, some of which are motivated by cryptographic applications.