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2001 Lester R. Ford Award Recipient

Greg Martin, University of British Columbia

The Lester R. Ford Awards, established in 1964, are made to authors of expository articles published in The American Mathematical Monthly. The Awards are named for Lester R. Ford, Sr., a distinguished mathematician, editor of The American Mathematical Monthly, 1942-46, and President of the Mathematical Association of America, 1947-48.

Normal numbers are those whose decimal (or other base) expansions contain all possible finite configurations of digits with roughly their expected frequencies. It is known that almost all real numbers are normal to any given base. Thus, the set of absolutely abnormal numbers, those that are not normal to any base, must be a set of measure zero. The rational numbers provide easy examples of such numbers. However, the set of absolutely abnormal numbers is an uncountable dense subset of the real numbers, hence must contain irrational numbers. This article is unique in that the author actually constructs an example of an irrational absolutely abnormal number. In an exceptionally clear, straightforward fashion the reader is shown the motivation for considering Liouville numbers, followed by the description of one particular such number, and finally the main result, that this number is absolutely abnormal. Moreover, as pointed out by the author, this technique can be generalized to give uncountably many irrational absolutely abnormal numbers. This paper provides some very satisfying results in an area of mathematics where one can prove the existence of some types of numbers but actual constructions are rare.