Coming from a completely different world, I am trying to learn some very basic Fourier analysis, and have been scratching my head around this (it may be a very stupid question with an obvious answer):

Let $(f_n(t))_{n\in\mathbb{N}}$ be defined as $f_n(t)=\exp(it\cdot x_n)$ on some compact $T\subset\mathbb R^d$ for a given sequence $(x_n)_{n\in\mathbb N}$ in $\mathbb R^d$. Is it possible to choose $T$ and construct a measure $\lambda$ on it in such a way to make $(f_n(t))_{n\in\mathbb{N}}$ an orthonormal basis in $L_2(T,\lambda)$? This would be in analogy with the usual Fourier basis on the unit circle where $d=1$ and $x_n$ are the integers. So, we need that $\int\exp(it\cdot(x_n-x_m))d\lambda(t)=0$, for $n\neq m$, which seems impossible for generic $(x_n)_{n\in\mathbb N}$, but it is not obvious that it is since we can choose T and the measure on it freely?

Many thanks, P.S.