On dispersion and Markov constants

Abstract

and put D({x,})=lim supN NdN. In particular he carried out investigations of sequences of the form x ,=n0(mod l) for irrational O's. For such a 0 he defined the dispersion constant by D(O)=D({nO(mod 1)}). It turns out that D(0)<o~ if and only if the continued fraction expansion of 0 has bounded partial quotients. Moreover if 01 and 02 are equivalent, then D(01)=D(0~). (Two numbers are called equivalent if their continued fraction expansions coincide from some point on.) He also shows that if 3 is equivalent to 01=(1 +(5-)/2 then D(0)=(5+3 [~) /10= =1.170..., if 0 is equivalent to ~2=]/2then D(~)=(1+~-2)/2=1.207... and if 0 is not equivalent to either 01 or 02 then D(O)~3-V3-=l .257 .... Thus the dispersion spectrum D, i.e. the set of all possible values of D(0), contains gaps. Niederreiter also identifies another gap in D: ((1 + ]/3)/2, (13 + 7 ]/]'3)/26) = (1.365..., 1.470...). All this suggests an analogy with Lagrange (or Markov) constants and Lagrange spectrum. For each irrational 3 the Lagrange (or Markov) constant is defined by