  We show that almost all natural numbers n not divisible by 4, and not
congruent to 7 modulo 8, are represented as the sum of three squares, one of
which is the square of an integer no larger than (log n)^{1+e} (any e>0). This
answers a conjecture of Linnik for almost all natural numbers, and sharpens a
conclusion of Bourgain, Rudnick and Sarnak concerning nearest neighbour
distances between normalised integral points on the sphere.
