Description
Title: A function field analogue of a theorem of Sarkozy, due to B Green.
Speaker: Niranjan Balachandran
Date-Time: Wednesday, February 28 2018, 2 PM to 3.30 PM
Venue: Ramanujan Hall
Abstract: In the late 70s Sarkozy proved the following theorem: Given a
polynomial f(T) over the integers with f(0)=0, there exists a constant c_f
such that for any set $A\subset [n]$ of size at least $n/(log n)^{c_f}$
there exist distinct $a,b\in A$ such that $a-b=f(x)$ for some $x$. In
2016, Ben Green proved a function field analog of the same result but with
a much better bound for $|A|$: Given a polynomial $F\in\bF_q[T]$ of degree
$k$ with $F(0)=0$, there exists $0 q^{(1-c)n}$
there exist $\alpha(T)\neq\beta(T)$ in $A$ such that
$\alpha(T)-\beta(T)=F(\gamma(T))$ for some $\gamma(T)\in\bF_q[T]$. We will
see a proof of this result.