It is well-known that the Hilbert space $L^2(\Omega)$ does not admit any orthogonal Fourier basis for the unit ball $\Omega=B_d\subset \Bbb R^d$, $d>1$. Motivated by this, we introduce the concept of $\phi$-approximate orthogonality. More precisely, given a bounded domain $\Omega$, and a bounded measurable function $\phi:[0,\infty) \to [0, \infty)$ with $\phi(t)\to 0$ as $t\to \infty$, we say that the functions $e_a(x):=e^{2\pi i x\cdot a}$ and $e_{a'}(x):=e^{2\pi i x\cdot a'}$, $a\neq a'$, are $\phi$-approximately orthogonal if

$$|\widehat{1_\Omega}(a-a')|\leq \phi(|a-a|).$$
As a result, any two orthogonal exponential functions are $\phi$-approximately orthogonal if we take $\phi\equiv 0$.

In this talk, we show that if $\phi$ decays faster than $(1+t)^{-\frac{d+1}{2}}$ as $t\to \infty$, then there is no set $A$ with positive and finite upper density such that the exponentials $\mathcal E(A):= \{e_a: a\in A\}$ are mutually $\phi$-approximately orthogonal on the ball. As a result, we show that for such $\phi$, the space $L^2(\Omega)$ does not admit any $\phi$-approximately orthogonal non-harmonic Fourier frame (frames of exponential forms).