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By Georg Cantor

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Extra resources for Contributions to the founding of the theory of transfinite numbers

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20. Find an English word containing two distinct squares, each of length ≥ 4. 21. Is the decimal√expansion of π squarefree? Cubefree? How about the decimal expansion of 2, or e? 22. Let u(n) denote the number of overlap-free binary words of length n. Show that u(n) is bounded by a polynomial in n. 23. Let w = c0 c1 c2 · · · = 0010011010010110011 · · · be the infinite word defined by cn = the number of 0’s (mod 2) in the binary expansion of n. Show that w is overlap-free. 24. Show how to extend the Thue–Morse infinite word t on the left to a two-sided infinite word that is still overlap-free.

Morse [1921] introduced the sequence t independently, as did Mahler [1927] and the Dutch chess master Max Euwe [1929]; see Exercise 18. ) Arshon [1937] introduced the Thue–Morse sequence in a slightly disguised form (as a fixed point of the map 1 → 12, 2 → 21) and observed that it was cubefree. Gardner [1961a, 1961b] popularized the problem of finding an infinite squarefree string in his Mathematical Games column; the columns were reprinted in Gardner [1967a, pp. 32–33, 90–95]. Noland [1962] proposed the problem of finding an infinite squarefree word over three symbols, and a solution was given by Braunholtz [1963].

Show how to generate a random primitive word in expected linear time. 47. Let k ≥ 2 be an integer. Show that the set of all subwords of the primes expressed in base k is {0, 1, . . , k − 1}∗ . 48. Let µ : 0 → 01, 1 → 10 be the Thue–Morse morphism. (a) Show that the lexicographically least infinite overlap-free word over 2 = {0, 1} starting with 1 is µω (1). (b) Show that the lexicographically least infinite overlap-free word starting with 0 is 001001ϕ ω (1). (c) What is the lexicographically least infinite overlap-free word over 2 that can be extended to a two-sided infinite overlap-free word?

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