= Bracket-linear frequency function
{title2=$\theta(h)=\beta\{\gamma h\}\pmod1$}
A frequency function built from an affine variable and its <fractional part>. Irrational $\gamma$ and $\beta$ can give many exact additive relations because each pair sum has only two possible floor carries. For example $\beta=\sqrt3$, $\gamma=\sqrt2$ gives $\gg N^3$ additive quadruples on $[-N,N]$, but agreement with every ordinary affine function occurs on only $o(N)$ shifts. Integer floor points underlying affine agreement must be collinear.
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