2023/05/01 by Rajula Srivastava, Srivastava, Rajula, Niclas Technau +1 · 1 citation
Engineering · Mathematics · #11D75 #11J83 #42B99 #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2305.01047
openalex publication_date 2023/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
For n≥ 3, let \mathscrM ⊆ℝn be a compact hypersurface, parametrized by a homogeneous function of degree d∈ ℝ>1, with non-vanishing curvature away from the origin. Consider the number N_\mathscrM(δ,Q) of rationals a/q, with denominator q∈ [Q,2Q) and a ∈ ℤn-1, lying at a distance at most δ/q from \mathscrM. This manuscript provides essentially sharp estimates for N_\mathscrM(δ,Q) throughout the range δ∈ (Qε-1,1/2) for d>1+\tfrac12n-3. Our result is a first of its kind for hypersurfaces with vanishing Gaussian curvature (d>2) and those which are rough (meaning not even C2 at the origin which happens when d<2). An interesting outcome of our investigation is the understanding of a `geometric' term (δ/Q)(n-1)/dQn (stemming from a so-called Knapp cap), arising in addition to the usual probabilistic term δQn; the sum of these terms determines the size of N_\mathscrM(δ,Q) for δ∈(Qε-1,1/2). Consequences of our result concern the metric theory of Diophantine approximation on `rough' hypersurfaces -- going beyond the recent break-through of Beresnevich and L. Yang. Further, we establish smooth extensions of Serre's dimension growth conjecture.