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Weird AoC

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The Axiom of Choice (AoC), while very much intuitive in its basic definition, can allow us to define weird functions and objects. We present a small list here. We will not talk about the "splitting a sphere in two" result as it way too overpopular for the wrong reasons (what makes the result special is all about only using rotations since by cardinality alone it's obvious that $S^3 \leftrightarrow S^3 \cup S^3$)

Constructing polynomials from periodic functions

Here is the problem: do there exist two periodic functions $f,g : \mathbb{R} \to \mathbb{R}$ such that $\forall x \in \mathbb{R},\ (f+g)(x)=x$?
Surprisingly, the answer is yes, but only if we allow very wild, non-continuous functions.

The identity

Choose an irrational number, say $\alpha = \sqrt 2$. Using a Hamel basis of $\mathbb{R}$ over $\mathbb{Q}$ containing both $1$ and $\alpha$, every real number $x$ can be written uniquely as $$ x = q \cdot 1 + r\alpha + \text{other basis terms}, $$ with $q,r \in \mathbb{Q}$. Then define $f(x)$ to be the $\alpha$-component and $g$ as follows: $$ f(x)=r\alpha. \ g(x)=x-f(x). $$
By construction, $f(x)+g(x)=x$. Moreover, adding $1$ to $x$ does not change the coefficient of $\alpha$, so $f$ is $1$-periodic. Similarly, adding $\alpha$ changes $f(x)$ by exactly $\alpha$, so $g(x)=x-f(x)$ is $\alpha$-periodic.
Thus the identity function is indeed the sum of two periodic functions.

Polynomials

More generally, every polynomial $P : \mathbb{R} \to \mathbb{R}$ of degree at most $d$ is the sum of $d+1$ periodic functions.
The idea is to choose real numbers $\alpha_0,\dots,\alpha_d$ that are linearly independent over $\mathbb{Z}$. Then every orbit of the group
$$ \alpha_0\mathbb{Z}+\cdots+\alpha_d\mathbb{Z} $$

looks like a copy of $\mathbb{Z}^{d+1}$. On such an orbit, the polynomial $P$ becomes a polynomial in $d+1$ integer variables.

But every monomial of total degree at most $d$ must miss at least one of these $d+1$ variables. So we split the monomials according to a variable they do not depend on. The part that does not depend on the $i$-th variable gives a function periodic with period $\alpha_i$.

Hence

$$ P = f_0+\cdots+f_d, $$

where each $f_i$ is periodic.

Exponential?

This phenomenon is special to functions killed by sufficiently many finite differences. If $f_i$ has period $a_i$, then

$$ \Delta_{a_i} f_i = 0, $$

where

$$ \Delta_a F(x)=F(x+a)-F(x). $$

Therefore, if a function $F$ is a finite sum of periodic functions, then for some nonzero periods $a_1,\dots,a_n$ we must have

$$ \Delta_{a_1}\cdots\Delta_{a_n}F=0. $$

Polynomials satisfy this property: each finite difference lowers the degree. After enough differences, the polynomial disappears.

But the exponential does not. Indeed,

$$ \Delta_a e^x = e^{x+a}-e^x = (e^a-1)e^x. $$

Since $a\neq 0$, the factor $e^a-1$ is nonzero. Applying finitely many differences only multiplies $e^x$ by nonzero constants; it never becomes zero.

So $e^x$ cannot be written as a finite sum of periodic functions.

Conclusion

Polynomials can be built from periodic functions, but only by using highly non-regular functions. The identity already requires a non-continuous, non-measurable construction. The exponential, on the other hand, is too rigid: finite differences never kill it, so it cannot be a finite sum of periodic functions.