Lectures on Diophantine Approximations Part 1: g-adic by Kurt Mahler

By Kurt Mahler

Notre Dame Mathematical Lectures, No. 7.

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Additional info for Lectures on Diophantine Approximations Part 1: g-adic numbers and Roth's theorem

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For the explicit construction of such approximations a simple algorithm will be given in the next chapter. It is based on the continued fraction algorithm for real numbers and forms its natural extension to p-adic, g-adic, and g*-adic numbers. The deep theorem of Roth shows that these approximations cannot be too good in the case of an algebraic number a. The theorem is bestpossible and has many interesting consequences. The long and involved proof fills all the remaining chapters. Before starting with these investigations, it has perhaps some interest to collect certain general properties that are basic for the theory of Diophantine approximations.

P-1; for of = 0 all these coefficients are zero, while for a £ 0 it may be assumed that A(f) * 0 and then la In = P"f because |A(*)|D = 1. 36 LECTURES ON DIOPHANTINE APPROXIMATIONS All the developments for the g-adic number A studied in this section have one important property in common. , |a|pr, respectively. Of particular importance are those g-adic numbers A which satisfy the inequality and are called g-adic integers. Their g-adic series do not contain non-zero terms A(m)gm with negative exponents m.

On taking m=n+l, it is also obvious that Y am cannot be convergent m=l unless lim am = 0 (w); m—>°o but, just as in real analysis, this condition is not in general sufficient for convergence. There is, however, one important case when it is sufficient, viz. that when w(a) is Non-Archimedean. For now w(an+l + an+2 + ... -, w(am)]. and so, if {am} is & null sequence with respect to w(a), the righthand side is smaller than e f or all m > n In the following chapter these simple remarks on convergent series will be applied to series for p-adic, g-adic, and g*-adic numbers.

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