[Paper Review] Trigonometric approximation and a general form of the Erd\H{o}s Tur\'{a}n inequality

Leonardo Colzani, Giacomo Gigante+1 more|arXiv (Cornell University)|Jan 6, 2010

Mathematical Approximation and Integration10 references3 citations

TL;DR

This paper establishes a general framework for trigonometric approximation of characteristic functions of measurable sets using entire functions of exponential type, leading to sharp Erd\'os-Tur\'an-type discrepancy bounds for point distributions on the torus and compact Riemannian manifolds. The key contribution is a universal, geometry-independent function ψ(t) controlling the approximation error in terms of distance to the boundary, enabling optimal discrepancy estimates that extend classical results to arbitrary measurable sets and manifolds with explicit dependence on Minkowski content-like regularity parameters.

ABSTRACT

There exists a positive function $\psi(t)${on}$t\geq0${, with fast decay at infinity, such that for every measurable set}$\Omega${in the Euclidean space and}$R>0${, there exist entire functions}$A(x) ${and}$B(x) ${of exponential type}$R${, satisfying\}$A(x)\leq \chi_{\Omega}(x)\leq B(x)${and}$| B(x)-A(x)| \leqslant\psi(R\operatorname*{dist}(x,\partial\Omega)) $. This leads to Erd\H{o}s Tur\'{a}n estimates for discrepancy of point set distributions in the multi dimensional torus. Analogous results hold for approximations by eigenfunctions of differential operators and discrepancy on compact manifolds.

Motivation & Objective

• To extend the classical Erd\'os-Tur\'an inequality to arbitrary measurable sets in multidimensional tori and compact Riemannian manifolds, beyond intervals and smooth domains.
• To develop a universal approximation method for characteristic functions using entire functions of exponential type, independent of the set's regularity or geometry.
• To quantify the discrepancy of point distributions (lattices, arithmetic progressions, group orbits) in terms of geometric and spectral properties of the domain, particularly the Minkowski content of its boundary.
• To generalize the classical trigonometric polynomial approximation of intervals to eigenfunctions of the Laplace-Beltrami operator on manifolds, preserving error control via distance to the boundary.
• To achieve optimal or near-optimal bounds on discrepancy, with explicit dependence on the Minkowski content parameter δ and point set structure.

Proposed method

• Construct entire functions A(x) and B(x) of exponential type R such that A(x) ≤ χΩ(x) ≤ B(x) and |B(x) − A(x)| ≤ ψ(R dist(x, ∂Ω)), where ψ(t) decays rapidly at infinity.
• Use the kernel KR(x,y) = ∑λ h(R⁻¹λ) ϕλ(x)ϕλ(y) with h(|ξ|) = (1 + |ξ|²)⁻(d+1)/2 m∗m(ξ), ensuring asymptotic Euclidean behavior and controlled remainder.
• Define auxiliary functions HR(x) = β(1 + R dist(x, ∂Ω))⁻α to control the approximation error, and use them in integral representations of A(x) and B(x) via the kernel.
• Apply the Erd\'os-Tur\'an framework by bounding the discrepancy via the L¹ norm of HR and the spectral norms of the point distribution operator T.
• Leverage spectral theory of elliptic operators to bound ∑|ϕλ(p)|² ≤ cRᵈ and ∑|bχΩ(λ)|² ≤ cM(δ,Ω)2⁻δk for λ ≥ 2ᵏ.
• Use the eigenvalue decay ρ(m) of the averaging operator T to control the contribution of non-constant eigenfunctions, leading to bounds involving ρ(m) and M(δ,Ω).

Experimental results

Research questions

• RQ1 Can the classical Erd\'os-Tur\'an inequality be generalized to arbitrary measurable sets in the multidimensional torus, not just intervals?
• RQ2 What is the optimal dependence of the discrepancy bound on the geometric regularity of the set, measured via the Minkowski content parameter δ?
• RQ3 How can trigonometric approximation be extended from intervals to eigenfunctions of the Laplace-Beltrami operator on compact Riemannian manifolds?
• RQ4 Can point distributions generated by free group actions on homogeneous spaces achieve discrepancy comparable to lattice points, and what spectral conditions ensure this?
• RQ5 What is the sharp dependence of the discrepancy on the number of points m, the geometry of the domain, and the spectral properties of the point set?

Key findings

• For any measurable set Ω in the torus Tᵈ with M(α,Ω) < γ, the discrepancy of lattice points L(m) satisfies sup |μ(Ω) − m⁻¹∑χΩ(xⱼ)| ≤ cγm⁻α/d for some constant c.
• For almost every x ∈ Tᵈ, the discrepancy of arithmetic progressions {jx}ⱼ=1ᵐ is bounded by cγm⁻α/d log^(α(d+1+ε)/d)(m), improving classical bounds.
• For convex polyhedra with facets parallel to a fixed set X, there exists a lattice point g such that the discrepancy of {jg/m}ⱼ=1ᵐ is ≤ cm⁻¹logᵈ(m), matching the best known bounds in 2D.
• On compact manifolds, the discrepancy of group-orbit point sets satisfies sup |μ(Ω) − m⁻¹∑χΩ(σⱼx)| ≤ cM(δ,Ω)(R⁻δ + R⁽ᵈ⁻δ⁾/²ρ(m)) for any R > 0.
• For the 2-sphere with free group action, the discrepancy bound is cM(δ,Ω)m⁻δ/(2+δ)log²δ/(2+δ)(m), matching the best known result for spherical caps.
• The spectral eigenvalue decay ρ(m) ≤ cm⁻¹/²log(m) for SO(3)/SO(2) leads to the optimal m⁻δ/(2+δ) dependence in the discrepancy bound, confirming sharpness in the spherical case.