# Exponential polynomials which are equal in modulus.

Let $f(t)=\sum_{k=1}^n c_k e^{ia_k t}$ and $g(t)=\sum_{k=1}^n d_k e^{ib_k t}$ be exponential polynomials with complex coefficients and having $n$ terms; and suppose they are equal in modulus: $$|f(t)|=|g(t)| \qquad \text{for } t\in\mathbb{R}$$ What does this imply about their exponent sets $A:=\{a_k\}$ and $B:=\{b_k\}$? In particular, does it follow that $A-A=B-B$ and how would you prove this?

Here's a start: $|f(t)|^2=|g(t)|^2$ is equivalent to $$\sum_{k,j} c_k c_j^* e^{i(a_k - a_j)t} =\sum_{k,j} d_k d_j^* e^{i(b_k - b_j)t}$$

which can be written $$\sum_l C_l \cos(A_l t+ \phi_l) = \sum_l D_l \cos(B_l t+ \psi_l)$$ where the $A_l$ and $B_l$ are of the form $|a_j-a_k|$ and $|b_j-b_k|$ respectively, and appear only once in their respective sums. The desired result $A-A=B-B$ follows only if none of the coefficients $C_l$ and $D_l$ vanish.

Now, they can't all vanish of course, so the above argument at least implies the equality of some subsets of $A-A$ and $B-B$.

If $a_1,\ldots,a_n$ are distinct real numbers, then the functions $\exp(i a_1 t),\ldots,\exp(i a_n t)$ are linearly independent over $\mathbb R$.
This can be shown inductively. For $n=1$ it is obvious. Assume that it holds for $n=k$, and $a_1,\ldots,a_k,a_{k+1}$ are distinct, and $$c_1\exp(i a_1 t)+\cdots+c_k\exp(i a_k t) +c_{k+1}\exp(i a_{k+1} t)=0,$$ for all $t$, where $c_1,\ldots,c_{k+1}\in \mathbb R$. Then $$c_1\exp(i (a_1-a_{k+1}) t)+\cdots+c_k\exp(i (a_k-a_{k+1}) t) +c_{k+1}=0,$$ and differentiating we obtain $$i (a_1-a_{k+1})c_1\exp(i (a_1-a_{k+1}) t)+\cdots+i (a_k-a_{k+1})c_k\exp(i (a_k-a_{k+1}) t)=0,$$ which by the inductive hypothesis implies that $$i (a_1-a_{k+1})c_1=\cdots=i (a_k-a_{k+1})c_{k+1}=0,$$ and since $a_j\ne a_\ell$, then $$c_1=\cdots=c_k=0,$$ and hence $c_{k+1}=0$, as well.
• It can be used with the equation $$\sum_{k,j} c_k \bar c_j \mathrm{e}^{i(a_k - a_j)t} =\sum_{k,j} d_k \bar d_j \mathrm{e}^{i(b_k - b_j)t}.$$ – Yiorgos S. Smyrlis Apr 2 '14 at 21:30
• Okay, group like terms and rewrite this equation as $$\sum_{l}C_l e^{i A_l t}= \sum_{l}D_l e^{i B_l t}=$$ where $$C_l = \sum_{k,j:a_k-a_j=A_l} c_k c_j^*$$ and $$D_l = \sum_{k,j:b_k-b_j=B_l} d_k d_j^*$$ Linear independence implies that sets $\{a_k-a_j: C_l \neq 0\}=\{b_k-b_j: D_l \neq 0 \}$, but this is less than the desired result $A-A=B-B$ in the notation above. Please elaborate. – MathManM Apr 2 '14 at 21:41