I've been trying to solve this inequality but I only get the obvious part which is the $\leq$ part. I need the $<$. The problem is the following:
Given a subset $A\subset [0,1)$ of measure not $0$, prove the following strict inequality: $$ |\widehat{\chi_A}(n)| = \bigg|\int_A e^{2\pi i n x} d x \bigg| < |A| $$
where $\widehat{\chi_A}(n)$ denotes the $n$-th Fourier coefficient of the function $\chi_A$, that is, $1$ in $A$ and $0$ outside.
Thank you very much.
Let $f(x)=e^{2\pi inx}$. If equality holds then $\int_A f(x)dx=|A|e^{it}$ for some real $t$. Hence $\int_A [e^{-it}f(x)-1]dx=0$. Taking real part we get $\int_A [\cos (2\pi nx-t)-1]dx=0$. The integrand is $\leq 0$ so this implies $\cos (2\pi nx-t)-1=0$ for almost all $x \in A$. In other words $2\pi nx-t \in 2\pi \mathbb Z$ for almost all $x$. can you get a contradiction from this? [Hint: take $x,x'$ with $x-x'$ irrational]. Works only if $n \neq 0$. Thanks to David Ullrich for pointing out.