Periodic triangular bump ($b_{\epsilon }(t)=(1-\Vert t\Vert /\epsilon {)}_{+}$)

For $0<\epsilon \le 1/2$, this continuous function on $\mathbb{R}/\mathbb{Z}$ has nonnegative Fourier coefficientsIt is ${\epsilon }^{-1}$ times the periodic convolution of two indicators of $[-\epsilon /2,\epsilon /2]$. Its Fourier series is absolutely convergent, ${\sum }_r{\hat{b}}_{\epsilon }(r)=b_{\epsilon }(0)=1$, and ${\sum }_{|r|>R}{\hat{b}}_{\epsilon }(r)\ll 1/(\epsilon R)$. Consequently, if all $N$ points $t_n$ avoid the interval $\Vert t\Vert <\epsilon$, some frequency $1\le r\ll {\epsilon }^{-2}$ satisfies $|{\sum }_{n}e(r{t}_n)|\gg N\epsilon$. This detects missing small neighborhoods by a large exponential sum.