When you see the word “almost” in a mathematical context, it might be used informally, but often it has a precise meaning. I wrote about this before in the post Common words that have a technical meaning.

Often the technical meaning of “almost” is “within any finite tolerance.” That’s how it is used in the context of **almost periodic functions**.

A function *f* is periodic with period *T* if for all *x*,

*f*(*x* + *T*) = *f*(*x*)

For example, the sine function is periodic with period 2π.

So what does **almost periodic** mean? It means that for any ε > 0, there exists a *T* > 0 such that

|*f*(*x* + *T*) – *f*(*x*)| < ε

for all *x*. Note that the value of *T* depends on the value of ε. You tell me your tolerance for “almost” and in theory I could hand you back *T* that meets your tolerance.

That’s in theory. Can we actually do it in practice? Let’s consider

*f*(*t*) = sin(2πα*t*) + sin(2πβ*t*)

where α/β is irrational. In the previous post we had α = 1/log 2 and β = 1/log 5.

The Hurwitz approximation theorem says that because α/β is irrational, there are infinitely many integers *p* and *q* such that

|α/β – *p*/*q*| < 1/(√5*q*²).

To put it another way, we can find *p* and *q* such that

|*q*α – *p*β| < β/√5*q*,

i.e. we can find *p* and *q* such that *q*α and *p*β are close together as we wish by looking for a large enough *q*, and Hurwitz promises us that we can find a *q* as large as we want.

If we let *T* = *q*/α then the first component of *f* is exactly periodic, i.e.

sin(2πα(*t* + *q*/α) = sin(2πα*t*).

With a little effort we can show that

sin(2πβ(*t* + *q*/α)) = sin(2πβ*t* + *2*π*(qβ – p*α)*t*/α)

and we said above we can make

*qβ – p*α

as small as we like by choosing a large enough *q* in Hurwitz’ theorem. And so we can choose *q* large enough that

sin(2πβ(*t* + *q*/α)) – sin(2πβ*t*)

is uniformly as small as we’d like.

One of the standard reference works for the number-theoretic aspects of almost periodic (as well as limit-periodic!) functions is the book “Arithmetical Functions” from Schwarz and Spilker: https://www.cambridge.org/core/books/arithmetical-functions/504B94E03437D4788E67D6EFCD2F9376

(also the chapters on Ramanujan Expansions are worth the read!)