Suppose a, b are two irrational numbers that we want to compare somehow. Here's one idea: Consider A_n = {k*a mod 1 | 1 <= k <= n} and B_n = {k*b mod 1 | 1 <= k <= n} to be subsets of the unit-circumference circle R/Z, which is an abelian group. Let the distance d(x, Y) between a point x of R/Z and a finite subset Y of R/Z be defined as d(x, Y) = the shortest arc xy of R/Z with y in the set Y. Now for any two finite subsets X, Y of of R/Z, define their distance dist(X, Y) as dist(X, Y) = max {min_{x in X} d(x, Y), min_{y in Y} d(X, y)}. Now make the distance rotation-invariant by changing the definition to Dist(X, Y) = min_{g, h in R/Z} dist(g + X, h + Y) or equivalently Dist(X, Y) = min_{g in R/Z} dist(g + X, Y). Finally compute f(n) = D(A_n, B_n) = Dist(A_n, B_n). QUESTION: --------- What is the asymptotic behavior of f(n) as n —> oo (for various irrationals a, b) ??? —Dan