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We prove that the space of continuous periodic functions is a set of first category in the space of almost periodic functions, and we also show that the space of almost periodic functions is a set of first category in the space of almost automorphic functions.

Since the last century, the study on almost periodic type functions and their applications to evolution equations has been of great interest for many mathematicians. There is a large literature on this topic. Several books are especially devoted to almost periodic type functions and their applications to differential equations and dynamical systems. For example, let us indicate the books of Amerio and Prouse [

Although almost periodic functions have a very wide range of applications now, it seems that giving an example of almost periodic (not periodic) functions is more difficult than giving an example of periodic functions. Also, there is a similar problem for almost automorphic functions. In this paper, we aim to compare the “amount” of almost periodic functions (not periodic) with the “amount” of continuous periodic functions, and we also discuss the related problems for almost automorphic functions.

Throughout the rest of this paper, we denote by

A function

Recall that

A function

Similar to the proof in [

A function

Recall that there exists an almost automorphic function which is not almost periodic, for instance, the following function:

Before the proof of our main results, we need to recall the notion about the first category.

Let

For

Then, it is easy to see that

Let

It suffices to prove that, for every

We denote

where

By Remark

In conclusion,

Since

Firstly,

By Theorem

H.-S. Ding acknowledges support from the NSF of China (11101192), the Chinese Ministry of Education (211090), the NSF of Jiangxi Province (20114BAB211002), and the Jiangxi Provincial Education Department (GJJ12173).