An article led me to generate some graphs of exponential sums of the form $S(N)=\sum_{n=0}^Ne^{2\pi i f(n)}$, where $f(n)= {n\over a}+{n^2\over b}+{n^3\over c}$ with $a,b,c\in\mathbb{N}_{>0},\,$ leaving me amazed at their great variety. Here are some samples:
The top four rows show graphs that appear to be cycles (some highly asymmetrical), whereas the bottom two rows show graphs that appear to be truncations of what would continue without bound in a fixed direction. All choices of $a,b,c\in\mathbb{N}_{>0},\,$ appear to give one of these two kinds of behavior!
Naturally, I wondered what structures occur in the more general case where $f$ is an arbitrary rational polynomial. Upon viewing the graphs for hundreds of rational polynomials of various degrees with "random" coefficients, I'm struck by the seeming fact that they all have the following (surprising?) property:
The "walk" in the complex plane first visits a finite set of (say, $p$) points, then visits the same set of points except displaced by a constant amount $c\in\mathbb{C}$, then visits the same set displaced by $2c$, then by $3c$, etc. (If $c=0$, the walk is a cycle with period $p$, else it consists of a continually displaced copy of the initial set of $p$ points.)
In other words, given $f$, there exists $p\in\mathbb{N}_{>0}$ such that $S(n+p) – S(n) = S(p)-S(0)\ \ (= \text{constant $c$})$ for all $n\in\mathbb{N}.$
But why should it be that (apparently) every rational polynomial produces this behavior? This seems preposterous, but I haven't found a single counterexample.
Letting $e_n:= \exp(2\pi i f(n)),$ we can restate the conjecture as follows:
Conjecture: If $f:\mathbb{N}\to\mathbb{Q}$ is a polynomial with rational coefficients, then there exists a least integer $p>0$ such that $$e_n+e_{n+1}+\cdots+e_{n+p-1}=\text{constant ($c$, say)}\ \ \text{for all $n\in\mathbb{N};$}$$ that is, every block of $p$ consecutive terms in the sequence $\left(e_n\right )_{n\in\mathbb{N}}$ has the same sum!
Question 1: Is this a well-known fact? In any case, how to prove it? (Or counterexample, if it turns out to not be true?)
Question 2: Is it possible to determine, as a function of the polynomial coefficients, the number $p$ and constant $c$ (or at least whether the graph is cyclic)? (I mean, of course, some way simpler than computing the graph!)
EDIT:
As a general rule, it appears that $p$ always divides the product of the coefficient denominators (and is very often equal to that product), but I have no idea how to prove it.
As far as the constant "block sum" $c$ is concerned, searching has turned up the quadratic Gauss sum formula, which implies that if $f(n)={a\over b}n^2$ with $b$ an odd prime and $a$ an integer, then
$$c = \left(\frac{a}{b}\right)\cdot\begin{cases}
\sqrt{b} & \text{if}\ b\equiv 1\pmod 4, \\
i\sqrt{b} & \text{if}\ b\equiv 3\pmod 4
\end{cases}
$$
where the Legendre symbol is defined by
$$\left(\frac{a}{b}\right) =
\begin{cases}
1 & \text{if } a \text{ is a quadratic residue modulo } b \text{ and } a \not\equiv 0\pmod b, \\
-1 & \text{if } a \text{ is a non-quadratic residue modulo } b, \\
0 & \text{if } a \equiv 0 \pmod b.
\end{cases}$$
Are any similar formulas available when $f$ is more complicated than such a simple quadratic monomial?
Best Answer
Write $f(x)=g(x)/m$ where $g$ has integer coordinates. Then $$\exp(2\pi i f(n))=\zeta^{g(n)}$$ where $\zeta=\exp(2\pi i/m)$. Therefore $e_n$ repeats with period $m$, and any block of $m$ consecutive $e_n$ has the same sum.