By B.M.M. de Weger

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**Sample text**

K put and Q kWl l (decimal) be a natural digits. For (j) lW(k-j)] , = [Q /10 i i Q (j) i and define J by (j+1) l (j) + J(j) . = 10 WQ i i i Q Thus, the relevant (j) are blocks of l i j the n * n matrices J & 1 | . o . | . o Aj = | 1 | (j) 7 Q1 ... Q(j) n-1 & | E = | | 7 1 . o * | | | , | (j) Q 8 n * | | . l | 10 8 o . consecutive digits of 1 49 Q i . Define for the & | | o Dj = | | (j) 7 J1 ... J(j) n * | | | , | 8 Then it follows at once that Aj+1 EWAj = + Dj . (k) Q = Q . Put U = I , B = A .

Heuristics (cf. 3) tell us that in a generic case -n we expect |L| = X . We now can prove easily the following useful lemma. 7. Let X be a positive number such that 1 ( 2 2) > r (n+1) +(n-1)Wg WX l(G) 9 0 1 . 1) has no solutions with 1 -----Wlog(gWCWc/X ) < X < X . d 1 1 Remark. 22) X = X . 21) then fails, 1 0 C . 22) yields a reduced lower bound for 0 Proof. , x 1 n the lattice point ~ L of size 0 < X < X 1 log X 0 . 1) with X as above. Then n-1 2 2 ~2 2 2 ~2 = g W S x + L < (n-1)Wg WX + L , i 1 i=1 2 |x| and ~ |L-gWCWL| < which is < nWX 1 n n S |x |W|[gWCWy ]-gWCWy | < S |x | , i i i i i=1 i=1 .

13 on certain sublattices that are useful for our applications. 2. Homogeneous one-dimensional approximation in the real case: continued fractions. We first study the case L = x Wy + x Wy . 1 1 2 2 Put y = -y /y . We assume that 1 2 fraction expansion of y be given by y is irrational. Let the continued y = [ a , a , a , .... ] , 0 1 2 and let the convergents p /q n n for n = 0, 1, 2, ... 37 be defined by & p-1 = 1 , { 7 q-1 = 0 , p = a q = 0 , p = a Wp + p 1 , q = a Wq + q 0 0 n+1 n+1 n+1 n n+1 n n-1 .