By Raymond Ayoub
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Additional resources for An introduction to the analytic theory of numbers
Xd−1 | ≤ H. Let λi = λi (H ), i = 1, . . , d, be the successive minima of C(H ) (see Ap(i) (i) pendix B for the deﬁnition). We denote by (x0 , . . , xd−1 ), i = 1, . . , d, 34 Approximation to algebraic numbers linearly independent points at which the successive minima λi are attained and we set (i) (i) (i) Pi (X ) = x0 + x1 X + . . + xd−1 X d−1 . We ﬁrst show that there exists some integer i for which |Pi (ξ )| ≥ λi H d! 7) holds. To this end, we set M = max 1≤i≤d |Pi (ξ )| , λi and we observe that λ1 , .
For g > 1, an easy induction on n yields that Jn (g) = 1 g n−1 i=0 (log g)i . i! 10) that λ E n (e An ) < 2n e−An n−1 i=0 (An)i (An)n ≤ 2n e−An n < 2n e−An n(Ae)n . i! n! 2 that almost all real numbers ξ in [0, 1] belong to only a ﬁnite number of sets E n (e An ). In other words, for almost all ξ in [0, 1] and for any sufﬁciently large integer n (in terms of ξ ), we have a1 . . 3, qn < 2an qn−1 < . . < 2n e An . Since for any real number ξ the denominators of the convergents of ξ and ξ +1 are the same, the theorem is proved.
4. 1. 13) with c = 1. Show that this result holds true for a real number ξ with a constant c strictly less than 1 if, and only if, ξ is badly approximable. A stronger statement is due to Davenport and Schmidt : Let ξ = [a0 ; a1 , a2 , . . ] be a real number, and set c5 (ξ ) = lim inf [0; an , an−1 , . . , a1 ] × [0; an+1 , an+2 , . . ]. 13) has a solution for all Q sufﬁciently large, and this is not true if c < 1/(1 + c5 (ξ )). 7 √ to prove this result and show that the largest possible value for c5 (ξ ) is (3 − 5)/2.
An introduction to the analytic theory of numbers by Raymond Ayoub