Effective equidistribution of Galois orbits for mildly regular test functions

In this paper we provide a detailed study on effective versions of the celebrated Bilu’s equidistribution theorem for Galois orbits of sequences of points of small height in the $N$-dimensional algebraic torus, identifying the quantitative dependence of the convergence in terms of the regularity of the test functions considered. We develop a general Fourier analysis framework that extends previous results obtained by Petsche (2005), and by D’Andrea, Narváez-Clauss and Sombra (2017).

💡 Research Summary

This paper investigates quantitative versions of Bilu’s equidistribution theorem for Galois orbits of points of small Weil height in the N‑dimensional algebraic torus (ℂ^×)^N. Bilu’s original result asserts that for a strict sequence {ξ_k}⊂(ℚ^×)^N with heights h(ξ_k)→0, the discrete measures supported on the Galois orbits converge weakly to the Haar measure on the unit poly‑circle (S^1)^N for every bounded continuous test function F. The authors aim to replace the vague “bounded continuous” hypothesis by milder regularity conditions and to give explicit rates of convergence in terms of the height.

Key definitions:

• For ξ∈(ℚ^×)^N, the Galois orbit S(ξ) is the finite set {σ(ξ) : σ∈Gal(ℚ̄/ℚ)}.
• The discrepancy of a test function F is E(F,ξ)=∫F dμ_{S(ξ)}−∫F dμ_{(S^1)^N}.
• The generalized degree D(ξ)=min_{n∈ℤ^N{0}}‖n‖_1·deg χ_n(ξ), where χ_n(ξ)=∏ ξ_i^{n_i}.
• The modified height h_D(ξ)=h(ξ)+log(2D(ξ))/D(ξ), which tends to zero along strict sequences.

The authors work in the logarithmic‑polar coordinates (θ,s)∈T^N×ℝ^N, identifying (ℂ^×)^N with T^N×ℝ^N via z_i=e^{2πiθ_i+s_i}. For a function F(θ,s) they consider its Fourier transform bF(n,t)=∫{T^N×ℝ^N}F(θ,s)e^{-2πi n·θ}e^{-2πi t·s} ds dθ, and the angular Fourier coefficients cF₀(n)=∫{T^N}F(θ,0)e^{-2πi n·θ} dθ.

Two complementary regularity frameworks are developed.

1. Fourier‑space regularity.
   Define the class A of test functions that are continuous, belong to L²(T^N×ℝ^N), and have bF∈L¹(ℤ^N×ℝ^N). For non‑decreasing functions G,H with G(x)/√x and H(x)/√x non‑increasing, Theorem 2 shows E(F,ξ) ≤ 2C₁(F,G)·G(8πh(ξ))^{-1}+C₂(F,H)·H(24h_D(ξ))^{-1}, where C₁(F,G)=∫{ℤ^N×ℝ^N}|bF(n,t)| G(‖t‖∞) dt and C₂(F,H)=∑_{n≠0}|cF₀(n)| H(‖n‖_1). Choosing G=H=x^γ with 0<γ≤½ yields Corollary 3: E(F,ξ) ≤ C(F)·h_D(ξ)^γ, with an explicit constant C(F). The exponent γ=½ is proved optimal (up to logarithmic factors) by constructing a function in A and a sequence of points attaining the lower bound.

If one only assumes bF∈L¹, Theorem 4 introduces the tail function ν_{bF}(y)=∫_{‖n‖1+‖t‖1>y}|bF(n,t)| dt and a slowly growing weight W satisfying W(x)/x→0. Then E(F,ξ) ≤ 2(√{8π}+√6)·h_D(ξ)·W(h_D(ξ))^{-1}‖bF‖{L¹}+3·ν{bF}(W(h_D(ξ))^{-1}), showing that convergence still holds without any smoothness, albeit with a weaker rate depending on W.

2. Physical‑space regularity (angular‑radial split).
   Assume F is continuous, satisfies a uniform modulus of continuity ω at s=0: |F(θ,s)−F(θ,0)| ≤ ω(|s|) for all θ, and that its angular Fourier coefficients satisfy a weighted L¹ condition as in Theorem 5. Then E(F,ξ) ≤ ω(2h(ξ)) + C₂(F,H)·H(24h_D(ξ))^{-1}. If ω(x)=L_γ(F)·x^γ (Hölder regularity) and H(x)=x^γ, Corollary 6 gives the same bound as before: E(F,ξ) ≤ (2γL_γ(F)+24γ∑_{n≠0}|cF₀(n)|‖n‖_1^γ)·h_D(ξ)^γ, again with optimal exponent γ≤½.

Theorem 7 treats the case where only cF₀∈L¹ is known, introducing a tail ν_{cF₀}(y)=∑_{‖n‖_1>y}|cF₀(n)| and obtaining a bound analogous to Theorem 4.

The paper compares these results with earlier works: Petsche (2005) obtained a rate h_D(ξ)^{1/3} in dimension one under a stronger L¹‑type Fourier condition; D’Andrea‑Narváez‑Clauss‑Sombra (2017) required Lipschitz continuity and achieved h_D(ξ)^{1/2} in any dimension. The present work matches the ½ exponent while allowing only Hölder‑½ regularity on the radial part and a fractional derivative of order ½ on the angular Fourier side, thus relaxing the hypotheses substantially. Moreover, the constants are expressed explicitly in terms of Fourier integrals, making the estimates amenable to computation.

Appendix A applies the method to characteristic functions of angular sectors, yielding a bound for multidimensional angular discrepancy in terms of h_D. Appendix B collects auxiliary lemmas from