Density of a finite subset ($\delta (A)=|A|/|X|$)

The density of $A$ in a nonempty finite ambient set $X$ is the proportion $|A|/|X|$.

For nonempty finite set $I$ and $A\subseteq I$, let $\alpha =|A|/|I|$ be the density of a finite subset. The balanced indicator function $f=1_A-\alpha 1_I$, extended by zero to an ambient finite group, has sum zero and vanishing zero Fourier coefficient. The sum of $f$ on a subset $P\subseteq I$ is $|A\cap P|-\alpha |P|$, measuring the excess density of a finite subset there.