For a natural number m and a set T of residues modulo m, \(| T| =n\), the discrete Fourier transform of T is defined by \(f(k)=\sum_{t\in T}e(tk/m)\). Standard probabilistic arguments show that a random set satisfies \(\| f\| =\max | f(k)| =O(\sqrt{n \log m})\), hence \(\| f\| =o(n)\) can be achieved if n/log \(m\to \infty\) arbitrary slowly (while \(\| f\| \gg n\) can be shown for \(n\ll \log m)\). In the paper, a set is constructed with slightly weaker properties, but still with \(\| f\| =o(n)\) and \(n\ll (\log m)^{1+\epsilon}\). This is applied to construct an essential component with \(\ll (\log x)^{3+\epsilon}\) elements up to x; the optimal result here is (log x)\({}^{1+\epsilon}\), due to the reviewer [Proc. Lond. Math. Soc., III. Ser. 54, 38-56 (1987; Zbl 0609.10042)].