I ran into the following phylosophical question when I was working out Roth's Theorem:
Let $A$ be a subset of $\{1,2,\dots, N\}$. We can associate $A$ as a subset of cyclic additive group $\mathbb{Z}_N$ because we have at our disposal Fourier transform of $f:\mathbb{Z}_N\to \mathbb{C}$ defined by $$\hat{f}(m):=N^{-1}\sum \limits_{x\in \mathbb{Z}_N}f(x)e\left(-\frac{xm}{N}\right), \text{where} \ e(x):=e^{2\pi i x}.$$ But in Roth's theorem we consider $f$ as a characteristic function of $A$, i.e. $f=\chi_A$.
Why we cannot apply Fourier transform directly to the set $\{1,2,\dots, N\}$, but not to the group $\mathbb{Z}_N$? Maybe the set $\{1,2,\dots,N\}$ has poor nature comparing to the group $\mathbb{Z}_N$?
Maybe this question sounds pointless, but would be thankful if somebody can give clear and persuasive answer!