On quantum modular forms of non-zero weights

We study functions $f$ on $\mathbb Q$ which statisfy a ``quantum modularity'' relation of the shape $$f(x+1)=f(x), \qquad f(x) - |x|^{-k} f(-1/x) = h(x)$$ where $h:\mathbb R_{\neq 0} \to \mathbb C$ is a function satisfying various regularity conditions. We study the case $\Re(k)\neq 0$. We prove the existence of a limiting function $f^*$ which extends continuously $f$ to $\mathbb R$ in some sense. This means in particular that in the $\Re(k)\neq0$ case the quantum modular form itself has to have at least a certain level of regularity. We deduce that the values $\{f(a/q), 1\leq a<q, (a, q)=1\}$, appropriately normalized, tend to equidistribute along the graph of $f^*$, and we prove that under natural hypotheses the limiting measure is diffuse. We apply these results to obtain limiting distributions of values and continuity results for several arithmetic functions known to satisfy the above quantum modularity: higher weight modular symbols associated to holomorphic cusp forms; Eichler integral associated to Maass forms; a function of Kontsevich and Zagier related to the Dedekind $\eta$-function; and generalized cotangent sums.