According to the uniqueness theorem in Fourier analysis, for every $f \in L^1[0,1]$ that satisfies $$ \int_0^1 f(t)e^{-2\pi i t n} \, dt = 0 $$ for every $n$, one has $f=0$. Now suppose that $(d_n)_{n \in \mathbb Z}$ is a sequence in $[0,1]$ that is dense in $[0,1]$, e.g., assume that is uniformly distributed. What can we say about $f$, if $$ \int_0^{d_n} f(t)e^{-2\pi i t n} \, dt = 0 $$ for every $n \in \mathbb Z$? If $f$ is supported on $[0,\frac 1 2]$ then it is reasonable to conclude that $f=0$ since in this case we would have $$ \int_0^\frac{1}{2} f(t)e^{-2\pi i t k_n} \, dt = 0 $$ for a sequence $(k_n) \subset \mathbb Z$ that has density $\frac{1}{2}$. But what if the support of $f$ is larger?
I'd be happy for any thoughts, ideas, or references on such problems.
