Why Bohr's definition of almost periodicity?

Question

Bohr in his Almost Periodic Functions book writes that the following naive approach to almost periodicity does not work, he phrases it being not a happy choice.

Specifically, let $\mathcal A$ denote the functions defined over $-\infty < x < +\infty$ such that for every $f(x) \in \mathcal A$ and all $\epsilon >0$ there is a translation number $\tau(\epsilon)$ such that $$|f(x+\tau(\epsilon))-f(x)|<\epsilon$$ holds uniformly in $-\infty <x < +\infty.$

Bohr writes that

[...] but it can even be shown, by constructing suitable examples, that this class does not even remain invariant under the ordinary operation of addition, i.e., in case $f(x)$ and $g(x)$ are two functions of this class their sum $f(x) + g(x)$ need not belong to this class.

Can somebody give me some examples for which $f(x), g(x) \in \mathcal A$ but $f(x)+g(x) \notin \mathcal A$?

@PaulFrost Bohr defines an almost periodic function so that the translation numbers are relatively dense in reals. This means that for every $\epsilon$ there is an $L(\epsilon)>0$ so that for all real number $a$ the interval $[a, a+L(\epsilon)]$ has a translation number $\tau_a(\epsilon)$. Hence, any Bohr almost periodic function, $\mathcal {AP},$ is an element of $\mathcal A.$ Naively, of course, I thought that those are the same but apparently not because if $f,g\in \mathcal {AP}$ then $f+g \in \mathcal {AP}$ as he proves it, and then $\mathcal {AP} \subset \mathcal {A}$ per Bohr's comment. — hyportnex, Dec 02 '25 at 23:47
this is an interesting problem and the answer is probably kind of ugly and complicated for the simple reason that $\mathcal A$ contains the space of uniformly continous functions (which of course contains the space of uap's in the sense of Bohr), and that is closed under addition, so counterxamples have to come from non-uniform continuous functions in $\mathcal A$; now this means that for example we cannot build them by uniform convergence from local pieces that are periodic or uap, so it's going to be messy — Conrad, Dec 03 '25 at 15:36
would try something like $f(x)=\sum \phi (x/n!)$ (or variants) where $\phi$ is periodic with period $1$ and $\phi(0)=0$ but $\phi'$ unbounded near $0$ so we avoid uniform continuity, while then $\phi(x+N!)-\phi(x)$ has only terms with $n<N$ which may be controlled so one can take $t(\epsilon)=N!$ for large enough $N$ and of course shift this by an irrational for $g$ but details are tricky; another idea is to use bump functions and do a similar shifting but then the edges get tricky while in the periodic case avoiding uniform continuity while still controlling the small index $n$ is the issue — Conrad, Dec 03 '25 at 15:43
for multiplication there is the trivial (unbounded) counterexample $x$ which is in $\mathcal A$ but $x^2$ is not (while Bohr uap are bounded and closed under multiplication) but for addition things are tricky since we need counterxamples where even though $t(\epsilon)$ works smaller numbers do not necessarily work for the given $\epsilon$ (uniform contiunuity gives precisely that all $|t|<t(\epsilon)$ work for at least a (small enough) $t(\epsilon)$ and that type of property leads to addition closure the way it leads for unfiormly continuous functions, so we have to avoid it — Conrad, Dec 03 '25 at 15:49
@Conrad this is exactly what confuses me as you wrote "...the answer is probably kind of ugly and complicated for the simple reason that $\mathcal A$ contains the space of uniformly continuous functions (which of course contains the space of UAP's in the sense of Bohr), and that is closed under addition, so counterexamples have to come from non-uniform continuous functions in $\mathcal A$," but then this implies that the "naive" and "unhappy" definition should work if restricted to uniformly bounded and continuous functions as what UAPs are. I am really confused now ... — hyportnex, Dec 04 '25 at 07:35
Depends on what you mean by naive definition should work, in other words what properties you want uap functions to have etc - IMHO the most important one is (unique) Fourier expansion and that doesn't hold for bounded uniformly continuous functions (see $e^{-x^2}$ which has all means zero); or put another way bounded uap functions satisfy the $t(\epsilon)$ only locally for small such in general and a uap should satisfy it for sequences going to infinity to be "like" a periodic function — Conrad, Dec 04 '25 at 13:20
Meant bounded uniform continuous satisfy the $t(\epsilon)$ property only for small $t(\epsilon)$ in general - again $e^{-x^2}$ is such due to its decay, while uap should behave "like" periodic functions so satisfy the property for some $t({\epsilon})\to \infty$ too; there are other classes like weak uap that contain $e^{-x^2}$ and are still larger than the class of bounded uniformly continuous functions and are of interest defined using weakening Bochner definition of uap with shift subsequences; then there are the Besicovitch spaces and so on - it all depends on what properties you look for — Conrad, Dec 04 '25 at 13:31
@Conrad by "naive" I meant the existence of translation numbers $\tau(\epsilon)$ for arbitrarily small $\epsilon>0$ but without the need of them being relatively dense. — hyportnex, Dec 04 '25 at 13:36
Afaik Bohr introduced uap originally to characterize Dirichlet series in the space of analytic functions on half planes say but then the subject developed by itself and Besicovitch, von Neumann, Bochner and others extended it massively — Conrad, Dec 04 '25 at 13:38
Again it depends on what you look for - bounded uniform continuously functions satisfy the "naive" definition but are far from being uap in an intuitive sense of being "like" periodic functions; Bohr's remark is that even if we require a sequence of $t(\epsilon)$ going to infinity that is not enough, you need rd or an equivalent — Conrad, Dec 04 '25 at 13:39
@Conrad you remember well, Bohr's book has a short appendix discussing the relationship between UAP and Dirichlet series. And indeed I called it "naive" because this was what I originally thought myself as what UAP was or should be but then Bohr had me reoriented and puzzled... — hyportnex, Dec 04 '25 at 13:40
Would recommend Besicovitch book btw as it has the Bohr stuff, his extensions (and many others) and the Dirichlet stuff in a very readable package — Conrad, Dec 04 '25 at 13:42

The Other Terry · Accepted Answer · 2025-12-07T18:20:32.853

Here's a sketch of an example. Some details need to be filled in. Let $k_i$ be a rapidly increasing sequence of positive integers (i.e., super-exponential). We will define a function with translation numbers $\tau(\frac{1}{n})\sim 2^{n-1} \Pi_{i=1}^{n} k_i$ for $n\geq 1$. Let $\tau_n=2^{n-1} \Pi_{i=1}^{n} k_i$. The function will be defined inductively. Start with the tent map written as $\phi_1(x)=2 \min{\{|x|,|1-x|\}}$ on $[0,1]$ (i.e., $k_1=1$). Suppose $\phi_{n-1}$ is defined on $[0,\tau_{n-1})$ for $n>1$. Let $c_n=\max{\{\phi_{n-1}(x): x\in [0,\tau_{n-1})\}}$. Define $\phi_{n}$ in the following way: $$\phi_n(x)=\phi_{n-1}(x)\ \ \mbox{for}\ x\in [0,\tau_{n-1}),$$ $$\phi_n(x) = \Big( \frac{c_n n^2+1}{c_nn^2}\Big) \phi_{n-1}(x-\tau_{n-1}),\ \ x\in [i\tau_{n-1},(i+1)\tau_{n-1}), i\in \{1,2,\ldots,k_n-1\},$$ $$\phi_n(x) = \Big( \frac{c_n n^2}{c_n n^2+1}\Big) \phi_{n-1}(x-\tau_{n-1}),\ \ x\in [i\tau_{n-1},(i+1)\tau_{n-1}), i\in \{k_n+1, k_n+2, \ldots ,2k_n-1\}.$$ Define $\phi_n(2k_n \tau_{n-1})=0$.

Define the function $\phi(x)=\lim_{n\to \infty} \phi_n(x)$ and let $\phi(-x)=-\phi(x)$ for $x>0$. This function is determined by the parameter sequence $(k_n)$ and $\phi_1$. Also, $\phi_1$ could be a different function such as $\phi_1(x)=\sin{(2\pi x)}$. It's clear that $\phi$ is continuous and also straightforward to show that $\phi$ is almost periodic according to Bohr's "unhappy criterion".

Given a sequence $k_n$ and $\phi_1$, define the mapping $\Phi((k_n),\phi_1)=\phi$ as above.
To construct two almost periodic functions $\phi$ and $\psi$, let $k_n$ be a sequence with super-exponential growth and define $m_n$ satisfying: $$\Big(\frac{\sqrt{k_{n+1}}\Pi_{i=1}^{n} k_i}{\Pi_{i=1}^{n} m_i}\Big)\leq m_{n+1}<\Big(\frac{\sqrt{k_{n+1}}\Pi_{i=1}^{n} k_i}{\Pi_{i=1}^{n} m_i}\Big)+1.$$ Thus, $2^{n} \sqrt{k_{n+1}}\Pi_{i=1}^{n}k_i \leq 2^{n}\Pi_{i=1}^{n+1} m_i = 2^{n} \Big( \sqrt{k_{n+1}}\Pi_{i=1}^{n}k_i + \Pi_{i=1}^{n} m_i\Big)$.
Let $\phi=\Phi((k_n),\phi_1)$ and $\psi=\Phi((m_n),\phi_1)$. The translation numbers for $\phi$ and $\psi$ will be sufficiently far apart and incompatible such that $\phi + \psi$ will not be almost periodic using Bohr's unhappy criterion.

Remark: I'm not sure there will be free time to work on this in the near future. This construction should work, but requires more elbow grease, especially to show that $\phi + \psi$ is not almost periodic.

Thank you for this very interesting example, it will take some time for me, admittedly, to digest it in detail. — hyportnex, Dec 06 '25 at 21:04
@hyportnex This is an interesting question. Reading a little bit about Harald Bohr, he won a silver medal in the Olympics playing football/soccer for the Danish team. It seems like this type of example should have been published somewhere. I'm near 100% sure the example above will work, but may need a little tweaking. The key idea is that the parameter sequence $k_n$ can grow arbitrarily fast which causes the translation numbers to be very sparse. — The Other Terry, Dec 07 '25 at 18:28

Why Bohr's definition of almost periodicity?

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