Somehow "Can a nonconstant continuous function on the reals ..." led to this question; don't ask me why: For any irrational number c, let G(c) be the subgroup of the circle group R/Z defined by G(c) = {nc mod 1 | n in Z} This defines a circular ordering on the members of G(c): For any three distinct elements x, y, z of G(c), the (circular) sequence (x,y,z) is either clockwise or counterclockwise. Since the definition of G(c) provides a 1-1 correspondence K in Z <—> Kc mod 1 in S^1, where C denotes the circle of radius 1 given by S^1 = R/2piZ, which we also identify via [t] in R/2piZ <—> exp(it) with the unit circle in the complex plane. Now we can can think of this circular ordering O_c: G(c)^3 - diag^ —> {+1, -1}, ((( where diag^ = {(x,y,z) in G(c)^3 | xx(z-y) + yy(x-z) + zz(y-x) = 0} is the superdiagonal. ))) as applying to the integers: O_c(K,L,M)) = +1, (Kc mod 1, Lc mod 1 , Mc mod 1) is counterclockwise in G(c); = -1, otherwise. QUESTION: Given two irrationals b, c, consider sense-preserving bijections h (i.e., h preserves the circular orderings of all distinct triples) h: G(b) —> G(c) and let the set of all such be denoted H+(b,c) = {h: G(b) —> G(c) | O_b(K,L,M) = O_c(h(K), h(L), h(M))} Does there exist a real number p > 0 such that the inf over h in H+(b,c) of the expression disc(b,c) = Sum_{x in G(b)} |x - h(x)|^p is nonzero but finite? —Dan