Holomorphic almost modular forms

Holomorphic almost modular forms are holomorphic functions of the complex upper half plane that can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in SL(2, Z). It is proved that such functions have a rotation-invariant limit distribution when the argument approaches the real axis. AD example of a holomorphic almost modular form is the logarithm of Pi(n=1)(infinity) (1 - exp (2pi i n(2)z)). The paper is motivated by the author's previous studies [Int. Math. Res. Not. 39 (2003) 2131-2151] on the connection between almost modular functions and the distribution of the sequence n(2)x modulo one.