Disjunctive !-Words and Real Numbers

An !-word p over a nite alphabet is called disjunctive if every nite word over occurs as a subword in p. A real number is called disjunctive to base a if it has a disjunctive a-adic expansion. For every pair of integers a; b 2 such that there exist numbers disjunctive to base a but not to base b we explicitly construct very simple examples of such numbers. General versions of the following results are proved. If (ni)i2! is a strictly increasing sequence of positive integers with ni+1 3i for in nitely many i then P 3 ni is disjunctive to base 2. The number P 2 i! i is disjunctive to base a if a is even and not a power of 2. The sum P 2 c i is disjunctive to base 6 if c 3 is odd.