An almost periodic function

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This post takes a more abstract view of the previous post. That post looked at the concrete question of whether a number ever has the same sine in radians as in degrees. The relation between radians and degrees is irrelevant except that π/180 is an irrational number.

Suppose α and β are two positive numbers such that α/β is irrational. In the previous post, α = 1 and β = π/180. Then the function

f(x) = sin(αx) − sin(βx)

is almost periodic: it is not periodic, but it comes close to being periodic, as close as you’d like provided you’re willing to look over a sufficiently long range of x‘s.

The identity

sin(αx) − sin(βx) = 2 cos((α + β)x/2) sin((α − β)x/2)

shows that f(x) is the product of two periodic functions but is not periodic itself. The periods of the cosine and sine above never coincide because the ratio of their frequencies is irrational.

The zeros of f are not periodic, though they can be divided into two subsequences that are periodic.

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John D. Cook, PhD

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