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# equidistribution on the unit circle of a particular sequence of finite subsets

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.