Note on almost-algebraic numbers
Harvey Cohn · Bulletin of the American Mathematical Society · 1946
Introduction.According to a theorem of J. Liouville, 2 if 6 is an algebraic number of degree n, then any approximation by rational, p/q, is of such a nature that (i) \e-p/q\zkr* for a positive constant k.Liouville constructed his transcendental numbers as the limit of special sequences of rationals, p/q, which violated condition (1) regardless of the values of k and n, as q->oo.Thus Liouville constructed almost-rational numbers.E. Maillet 3 likewise found a lower bound for 0-a where now 0 is approximated by the quadratic numbers, a.He then violated his lower bound by substituting for 0 the value of an almost periodic simple continued fraction and for a a quadratic number, namely a periodic simple continued fraction that 0 almost represented.Thus he constructed an almost-quadratic transcendental.It is an elementary matter to find a lower bound for 0-a, where we now approximate 0 by an algebraic number not necessarily rational or quadratic.We could then try several departures.We could, for example, try to construct almost-cubic or almost-biquadratic transcendentals. 4On the other hand, we could use a diagonal method, that is, we could consider the limit of a rapidly converging sequence of algebraic numbers whose degree becomes indefinite.For example, a root of a power series with rational coefficients is the limit of a sequence of (algebraic) roots of the partial sums, and the speed of convergence is regulated by the remainder.If the remainder is too small we find that the root of our power series can be approximated too closely by algebraic numbers of varying degrees, namely the roots of