= Periodic triangular bump
{title2=$b_\varepsilon(t)=(1-\|t\|/\varepsilon)_+$}
For $0<\varepsilon\le1/2$, this continuous function on $\mathbb R/\mathbb Z$ has nonnegative <Fourier coefficients>
$$
\widehat b_\varepsilon(r)=\varepsilon\left(\frac{\sin(\pi r\varepsilon)}{\pi r\varepsilon}\right)^2,
\qquad \widehat b_\varepsilon(0)=\varepsilon.
$$
It is $\varepsilon^{-1}$ times the periodic <convolution> of two indicators of $[-\varepsilon/2,\varepsilon/2]$. Its <Fourier series> is absolutely convergent, $\sum_r\widehat b_\varepsilon(r)=b_\varepsilon(0)=1$, and $\sum_{|r|>R}\widehat b_\varepsilon(r)\ll1/(\varepsilon R)$. Consequently, if all $N$ points $t_n$ avoid the interval $\|t\|<\varepsilon$, some frequency $1\le r\ll\varepsilon^{-2}$ satisfies $|\sum_ne(rt_n)|\gg N\varepsilon$. This detects missing small neighborhoods by a large <exponential sum>.
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