We give bounds on the number of distinct differences \( N_a-a\) as \( a\) varies over all elements of a given finite set \( A \subset (\mathbb R/\mathbb Z)^d,\ d\geq 1\) and \( Na\) is a nearest neighbour to \( a\) .

We discuss the mean values of multiplicative functions over function fields. In particular, we adapt the authors' new proof of Halasz's theorem on mean values to this simpler setting. Several of the technical difficulties that arise over the integers disappear in the function field setting, which helps bring out more clearly the main ideas of the proofs over number fields. We also obtain Lipschitz estimates showing the slow variation of mean values of multiplicative functions over function fields, which display some features that are not present in the integer situation.

David Dummit and Hershy Kisilevsky observed from calculation that the Legendre symbols \( (p/q)\) and \( (q/p)\) are unequal for rather more than a quarter of the pairs of odd primes \( p \) and \( q\) with \( pq\leq x\) , during some calculations. In fact almost \( 30 \% \) of the \( pq\)'s up to a million satisfy \( p\equiv q\equiv 3 \pmod 4\) . Together we found that this is no accident and that the bias up to \( x\) is roughly \( 1 +1/3(\log\log x-1)\) . This is a much stronger bias than the traditional "prime race" problem. When doing the math one finds that this problems about \( pq\) 's is equivalent to the prime race problem, for primes \( \equiv 3 \pmod 4\) versus those \( \equiv 1 \pmod 4\), in which we weight each prime by its reciprocal.

Here we survey the latest developments about sum-product estimates, especially over finite fields