Counting dyadic equipartitions of the unit square

Abstract

A dyadic interval is an interval of the form [ j/2 k ,( j+1)/2 k ], where j and k are integers, and a dyadic rectangle is a rectangle with sides parallel to the axes whose projections on the axes are dyadic intervals. Let u n count the number of ways of partitioning the unit square into 2 n dyadic rectangles, each of area 2 − n . One has u 0=1, u 1=2 and u n =2 u n−1 2− u n−2 4. This paper determines an asymptotic formula for a solution to this nonlinear recurrence for generic real initial conditions. For almost all real initial conditions there are real constants ω and β (depending on u 0, u 1) with ω>0 such that for all sufficiently large n one has the exact formula u n=ω 2 n g(βλ n), where λ=2 5 −4≈0.472 , and g(z)=∑ j=0 ∞ c jz j , in which c 0=(−1+ 5 )/2 , c 1=(2− 5 )/2 , all coefficients c j lie in the field Q( 5 ) , and the power series converges for | z|<0.16. These results apply to the initial conditions u 0=1, u 1=2 with ω≈1.845 and β≈0.480. The exact formula for u n then holds for all n⩾2. The proofs are based on an analysis of the holomorphic dynamics of iterating the rational function R( z)=2−1/ z 2.