Given a strictly convex function $g : [0, 1] \to \mathbb{R}$, I'm curious about the asymptotic distribution of the points $\exp{(2 \pi i N g(n / N))}$ for $n = 1, 2, \dots, N$, counted with multiplicity, as $N$ gets large. To be precise, I'm curious what sorts of bounds one can expect for the "discrepancy"

$$N^{-1}\#\{n : a < Ng(n/N) \pmod{1} < b\} - (b - a),$$

and in particular if these points become equidistributed as $N$ goes to infinity.

For what it's worth, I believe the answer is "yes." I have a guess about how to prove it, but I am not an analytic number theorist, and, even if my proof is right, I'm very curious if people know $1$) better ways to prove it, or $2$) preexisting results that imply it.

As for my humble stab at a sketch of a proof: First, I believe I have a proof in the case $g(x) = x^2$ using the Erd\"os-Tur\'an Inequality (here). (In this inequality, one has a bound on the above discrepancy which can shown to go to zero using what I understand are well-known formulas for quadratic Gauss sums.) Perhaps this is also true for arbitrary quadratic polynomials, and perhaps one can get good bounds on the discrepancy. For a general strictly convex function, $g$, I think it might be possible to approximate $g$ on small intervals (w.r.t. $N$) by a quadratic polynomial, and use the discrepancies computed in the polynomial cases to bound the discrepancy for $g$.

My proof for $g(x) = x^2$ will be provided only upon request: I am ashamed of it's ugliness. Thanks in advance.