Bracket-linear frequency function ($\theta (h)=\beta \{\gamma h\}(\mathrm{mod}1)$)

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 $\beta =\sqrt{3}$, $\gamma =\sqrt{2}$ 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.