On the existence of irregular solutions of the two-coefficient dilation equation (Q5945048)

The author proves the following two results. Assume that \(ab\neq 0\). If \(|a|>1\) or \(|b|>1\), then every nonzero compactly supported solution \(\varphi:\mathbb{R} \to\mathbb{R}\) of the functional equation \[ \varphi (x)=a \varphi (2x)+ b\varphi(2x-1)\tag{1} \] is unbounded in every neighbourhood of each point of \([0,1]\). For every real \(a\) and \(b\) such that \(ab\neq 0\) there exists a compactly supported solution \(\varphi:\mathbb{R}\to\mathbb{R}\) of (1) such that its graph meets every Borel subset of \([0,1]\times \mathbb{R}\) with uncountable vertical projection.