$L^p$-Fourier multipliers with bounded powers

We consider the space $M_p(\mathbb R^d)$ of $L^p$-Fourier multipliers and give a detailed proof of the following result announced by the authors in $\lbrack10\rbrack$: if $\varphi\colon\mathbb R^d\to \lbrack0, 2\pi\lbrack$ is a measurable function and $\|e^{in\varphi}\|_{M_p}=O(1)$, $n\in\mathbb Z$, for some $p\ne 2$, then the function $\varphi$ is linear in domains complementary to some closed set $E(\varphi)$ of Lebesgue measure zero, and the set of values of the gradient of $\varphi$ is finite. We also consider the question of which sets can appear as $E(\varphi)$. We study the behaviour of the norms of the exponential functions $e^{i\lambda\varphi}$ in the case when the frequency $\lambda$ tends to infinity along a sequence of real numbers. In particular, we construct a homeomorphism $\varphi$ of the line $\mathbb R$ which is non-linear on every interval and satisfies $\|e^{i2^n\varphi}\|_{M_p(\mathbb R)}=O(1)$, $n=0, 1, 2,\dots$, for all $p$, $1<p<\infty$.