A COMBINATORIAL METHOD FOR CONSTRUCTING NORMAL NUMBERS

Whereas most known methods of constructing normal numbers to a given i nteger base g greater-than-or-equal-to 2 consist in catenating a suita ble sequence of digit blocks and using its arithmetic or density prope rties, we introduce an elementary combinatorial method in order to sho w the normality of these real numbers. In this way simple proofs of th ese results arise. For a relatively wide class of ultimately positive real-valued functions f(x) with f(x) --> infinity as x --> infinity, d efined only in terms of growth, monotonicity and smoothness properties , it is proved that the number1 alpha(f):= alpha(g) (f) = . [\f(1)\] [ \f(2)\]... is normal. The following examples illustrate the kinds of f unctions f for which normality of the numbers alpha(g)(f) is establish ed here for the first time (p(n) = nth prime): a) f(n) = SIGMA(nu = 1) n gamma(nu)nbeta(nu) log(tau)(nu)n (d greater-than-or-equal-to 1; gamm a(d) > 0; beta(d) > beta(d - 1) > ... > 0; gamma(nu), delta(nu), tau(n u), is-an-element-of R), b) f(n) = (pin1/2 + n1/3)4/5/log2 n, c) f (n) = p(n)beta (beta > 0, beta is-an-element-of R), d) f(n) = p(p(n))2, e ) f(n) = n square-root i, (i) less-than-or-equal-to n < l(i + 1), i = 1, 2, for suitable l(1) < l(2) < ... On the negative side it is shown that no function of logarithmic order, such f(x) = log(tau)x, tau > 0, tau is-an-element-of R, produces a normal number alpha(f).