By Daniel J. Velleman

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We have to show that ti 2k−1 + 2k , which is equivalent to ti 2k−1 + 2k ≤ 614 c √ 2 ti 2k−2 + γi 2k−li −2 + 1 2 < ti 2k−1 + 2k + 1, THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 or in other words, √ √ √ √ 2 2 βi + γi − 2li +1 + 2 αi 2 2k−li −2 0≤2 βi − 2 2 √ √ 2 < 1. − αi 2 2k−li −1 + 2 k−li −1 Since γi − βi = 2αi and αi + βi = 2li +1 this is the same as √ √ √ 0 ≤ {αi 2 2k−li −1 } − 2 {αi 2 2k−li −2 } + √ 2 < 1. 2 (7) √ √ Relation (7) is true since 0 ≤ {x} − 2 {x/2} + 2/2 < √ 1 for all x ∈ R.

Niederreiter, Uniform Distribution of Sequences, John Wiley, New York, 1974. S. Rabinowitz and P. Gilbert, A nonlinear recurrence yielding binary digits, Math. Mag. 64 (1991) 168– 171. N. J. A. research. com/~njas/sequences. Th. Stoll, On families of nonlinear recurrences related to digits, J. Integer Seq. html. , On a problem of Graham and Erd˝os concerning digits, Acta Arith. 125 (2006) 89–100. D. from Graz University of Technology (Austria) in 2004. Currently, he is Maˆıtre de Conf´erences at the Universit´e de la M´editerran´ee, Marseille (France).

Moreover, the conclusions of Theorem 3 and Corollary 4, describing the form of the fundamental solution and the dependence of general solutions on the initial data at rational times, are also valid. By rescaling time, these results can be readily extended to the case when the coefficients of ω(k) are integer multiples of a common real number. 3 As always, the right-hand side really means the 2π-periodic extension of the indicated function defined on the interval 0 < x ≤ 2π. 2) whose dispersion relation is ω(k) = −k 2 .

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