Almost modular functions and the distribution of n2x modulo one

in the limit N → ∞. Interesting choices for ψ are as follows: (a) ψ(t) = χ[a,b](t), where χ[a,b] is the indicator function of the interval [a, b] + Z on S, with (b− a) ≤ 1; (b) ψ(t) = {t}, where {t} is the fractional part of t; (c) ψ(t) = e(t) := exp(2πit), leading to theta sums studied in [6, 7, 8, 14, 15]; (d) ψ(t) = log(1 − Ze(−t)), for some Z ∈ C, with |Z| = 1, and the sum in (1.1) becomes the logarithm of the polynomial

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