Intrinsic diophantine approximation on the unit circle and its lagrange spectrum

Let L(S-1) be the Lagrange spectrum arising from intrinsic Diophantine approximation on the unit circle S-1 by its rational points. We give a complete description of the structure of L(S-1) below its smallest accumulation point. To this end, we use digit expansions of points on S-1, which were originally introduced by Romik in 2008 as an analogue of simple continued fraction of a real number. We prove that the smallest accumulation point of L(S-1) is 2. Also we characterize the points on S-1 whose Lagrange numbers are less than 2 in terms of Romik's digit expansions. Our theorem is the analogue of the celebrated theorem of Markoff on badly approximable real numbers.