Holomorphic almost modular forms

Holomorphic almost modular forms are holomorphic functions of the complex upper half plane which can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in $\text{SL}(2,\text{ZZ})$. It is proved that such functions have a rotation-invariant limit distribution when the argument approaches the real axis. An example for a holomorphic almost modular form is the logarithm of ${\prod }_{n=1}^{\mathrm{\infty }}(1-\exp (2\pi \text{i}{n}^{2}z))$. The paper is motivated by the author’s studies [J. Marklof, Int. Math. Res. Not. {\bf 39} (2003) 2131-2151] on the connection between almost modular functions and the distribution of the sequence $n^{2}x$ modulo one.