Is there a result which gives a lower bound on the number of inverse pairs $(a, a^{-1})$ modulo prime $p$ lying in the interval $[1,t]$, where $t < p$?

The expected number of such pairs is about $t^2/p$. That the actual number is $t^2/p + O(p^{1/2} \log^2 p)$ follows from the upper bound $2 p^{1/2}$ on the size of a Kloosterman sum. In particular there's a nontrivial lower bound for $t \gg p^{3/4} \log p$. $$ $$ This bound (also for length-$t$ intervals mod $p$ other than $[1,t]$) appears as a Lemma in Merel's proof that for each $d$ there's an (explicit though large) upper bound on the size of the torsion group of an elliptic curve over any number field of degree $d$.
– Noam D. ElkiesJan 7 '13 at 1:51