ON DIOPHANTINE APPROXIMATION BELOW THE LAGRANGE CONSTANT Edward B. Burger and Jonathan M. Todd Department of Mathematics, Williams College, Williamstown, MA 01267 {Submitted May 1998-Final Revision April 1999)
1. INTRODUCTION For an irrational real number a, the Lagrange (often called the Markoff) constant for a, ju(a), is defined by ju(a) = limm£ q\\aq\\, q-*co
where || || denotes the distance to the nearest integer function (see [3], although there the Lagrange constant is defined to be //(a)" 1 ). Thus, for any c, 0 < cak+T-i\Hence, for each t9 0 < f < T 7 -1, we have PTn+t+k ~ at+kPTn+t+k-l
+
PTn+t+k-2
anC
^
^Tn+t+k
= a
f+fc
