VARIANCE OF NUMBER OF LATTICE POINTS IN RANDOM NARROW ELLIPTIC STRIP

Let N (x; s) be the number of integral points inside an elliptic strip of area s, bounded by the ellipses E (x) and E (x + s), where E (x) = {(q(1), q(2)) : (q(1)(2) + mu Q(2)(2)) pi mu(-1/2) = x}. We prove that if mu > 0 is a diophantine number and s = s (T) = const T-gamma, with 0 < gamma < 1/2, then [GRAPHICS] (with the factor 4 coming from symmetry considerations). Contrariwise, if mu is rational then [GRAPHICS] with some c (mu) > 0, and if mu is a Liouville number then [GRAPHICS] then [GRAPHICS] ely occupied by the dynamics restricted to this block. For the model in which 0's change to 1 when they have at least one neighboring 1 in each coordinate direction a further bound between exponential decay rates is also obtained, which in combination with results in [Mou] allows us to compute the exponential decay rate related to (i) for these models as being exactly -2log(1 - p), where p is the initial density of 1's. In particular the exponent nu related to (i) is equal to 1 for these models. This improves a result in [And].