Non-degenerate Curve Research Articles

Equidistribution of non-uniformly stretching translates of shrinking smooth curves and weighted Dirichlet approximation

AbstractConsider the action of $$a_t=\textrm{diag}(e^{nt},e^{-r_1(t)},\ldots ,e^{-r_n(t)})\in \textrm{SL}(n+1,{\mathbb {R}})$$ a t = diag ( e nt , e - r 1 ( t ) , … , e - r n ( t ) ) ∈ SL ( n + 1 , R ) , where $$r_i(t)\rightarrow \infty $$ r i ( t ) → ∞ for each i, on the space of unimodular lattices in $${\mathbb {R}}^{n+1}$$ R n + 1 . We show that $$a_t$$ a t -translates of segments of size $$e^{-t}$$ e - t about all except countably many points of a nondegenerate smooth horospherical curve get equidistributed in the space as $$t\rightarrow \infty $$ t → ∞ . This result implies that the weighted Dirichlet approximation theorem cannot be improved for almost all points on any nondegenerate $$C^{2n}$$ C 2 n curve in $${\mathbb {R}}^n$$ R n . These results extend the corresponding results for translates of fixed pieces of analytic curves due to Shah (2010) as well as those for uniform translates of shrinking curves due to Shah and Yang (2023), and answer some questions inspired by the work of Davenport and Schmidt (1969) and Kleinbock and Weiss (2008).

Winning of inhomogeneous bad for curves

AbstractWe prove the absolute winning property of weighted inhomogeneous badly approximable vectors on nondegenerate analytic curves. This answers a question by Beresnevich, Nesharim, and Yang. In particular, our result is an inhomogeneous version of the main result in (Duke Math J 171(14), 2022) by Beresnevich, Nesharim, and Yang. Also, the generality of the inhomogeneous part that we considered extends the previous result in An et al. (Adv Math 324:148–202, 2018). Moreover, our results even contribute to classical results, namely establishing the inhomogeneous Schmidt’s conjecture in arbitrary dimensions.

Lattice points on a curve via ℓ2 decoupling

This paper extends Bombieri and Pila’s estimate of lattice points on curves to arbitrary finite sets by incorporating considerations of minimal separation and the doubling constant. We derive the estimate by establishing the [Formula: see text] decoupling inequality for non-degenerate curves in [Formula: see text]. Additionally, we review the curve-lifting method introduced in Bombieri and Pila’s work and establish the estimate of lattice points in the neighborhood of a planar curve.

L^2 Estimates for a Nikodym Maximal Function Associated to Space Curves

For p in [2,infty ), we consider the L^p rightarrow L^p boundedness of a Nikodym maximal function associated to a one-parameter family of tubes in {mathbb {R}}^{d+1} whose directions are determined by a non-degenerate curve gamma in {mathbb {R}}^d. These operators arise in the analysis of maximal averages over space curves. The main theorem generalises the known results for d = 2 and d = 3 to general dimensions. The key ingredient is an induction scheme motivated by recent work of Ko-Lee-Oh.

An Exceptional Set Estimate for Restricted Projections to Lines in mathbb R^3

Let gamma :[0,1]rightarrow mathbb S^{2} be a non-degenerate curve in mathbb R^3, that is to say, det big (gamma (theta ),gamma '(theta ),gamma ''(theta )big )ne 0. For each theta in [0,1], let l_theta =text {span}(gamma (theta )) and rho _theta :mathbb R^3rightarrow l_theta be the orthogonal projections. We prove an exceptional set estimate. For any Borel set Asubset mathbb R^3 and 0le sle 1, define E_s(A):={theta in [0,1]: dim (rho _theta (A))<s}. We have dim (E_s(A))le max {0,1+frac{s-dim (A)}{2}}.

Sharp Sobolev regularity of restricted X-ray transforms

We study $L^p$-Sobolev regularity estimates for the restricted X-ray transforms generated by nondegenerate curves. Making use of the inductive strategy in the recent work by the authors, we establish the sharp $L^p$-regularity estimates for the restricted X-ray transforms in $\mathbb {R}^{d+1}$, $d\ge 3$. This extends the result due to Pramanik and Seeger in $\mathbb {R}^3$.

JARNÍK TYPE THEOREMS ON MANIFOLDS

AbstractLet $\psi $ be a decreasing function. We prove zero-infinity Hausdorff measure criteria for the set of dual $\psi $ -approximable points and for the set of inhomogeneous multiplicative $\psi $ -approximable points on nondegenerate planar curves. Our results extend theorems of Huang [‘Hausdorff theory of dual approximation on planar curves’, J. reine angew. Math.740 (2018), 63–76] and Beresnevich and Velani [‘A note on three problems in metric Diophantine approximation’, in: Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631 (American Mathematical Society, Providence, RI, 2015), 211–229] from s-Hausdorff measure, where $s\in \mathbb R$ , to the more general g-Hausdorff measure, where g is a suitable class of dimension functions.

Existence and reducibility of the Hilbert scheme of smooth and linearly normal curves in Pr of relatively high degrees

Let H d , g , r be the Hilbert scheme parametrizing smooth irreducible and non-degenerate curves of degree d and genus g in P r . We denote by H d , g , r L the union of those components of H d , g , r whose general element is linearly normal. In this article we show that H d , g , r L ( d ≥ g + r − 3 ) is non-empty in a certain optimal range of triples ( d , g , r ) and is empty outside the range. This settles the existence (or non-emptiness if one prefers) of the Hilbert scheme H d , g , r L of linearly normal curves of degree d and genus g in P r for g + r − 3 ≤ d ≤ g + r , r ≥ 3 . We also determine all the triples ( d , g , r ) with g + r − 3 ≤ d ≤ g + r for which H d , g , r L is reducible (or irreducible).

Sharp smoothing properties of averages over curves

Abstract We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve $\gamma $ in $\mathbb R^d$ , $d\ge 3$ . Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal $L^p$ Sobolev regularity estimates, which settle the conjecture raised by Beltran–Guo–Hickman–Seeger [1]. Besides, we show the sharp local smoothing estimates on a range of p for every $d\ge 3$ . As a result, we establish, for the first time, nontrivial $L^p$ boundedness of the maximal average over dilations of $\gamma $ for $d\ge 4$ .

Winning property of badly approximable points on curves

We prove that badly approximable points on any analytic nondegenerate curve in Rn are an absolute winning set. This confirms a key conjecture in the area stated by Badziahin and Velani in 2014 that represents a far-reaching generalization of Davenport’s problem from the 1960s. Among various consequences of our main result is a solution to Bugeaud’s problem on real numbers badly approximable by algebraic numbers of arbitrary degree. The proof relies on new ideas from fractal geometry and homogeneous dynamics.

A Sufficient Condition for a Curve to Belong to Boundary of its Own Convex Hullin An

The paper discusses some generalizations of the convex curve and the relationship between the classes of curves they define.&#x0D; The main result concerns the curves in n-dimensional affine space An: a non-degenerate curve in An lies on the boundary of its convex hull if any hyperplane intersects it in at most n points. It is justified by the theory of convex sets. This statement generalizes an earlier (2014) result of the author concerning the curves that are compact sets in An. V. Gustin proved in 1947 a stronger theorem for a Euclidean space En of dimension n: a connected set intersected by any hyperplane at no more than n points is a simple continuous curve lying on the boundary of a convex set. Note that in a projective space of dimension n the requirement that the curve intersects all planes at no more than n points is included in the definition of a convex curve. Thus, the established fact, together with the result of V. Gustin, shows the contiguity of the concepts of convexity in all three spaces.

On 𝐿¹² square root cancellation for exponential sums associated with nondegenerate curves in ℝ⁴

We prove sharpL12L^{12}estimates for exponential sums associated with nondegenerate curves inR4\mathbb {R}^4. We place Bourgain’s seminal result [J. Amer. Math. Soc. 30 (2017), pp. 205–224] in a larger framework that contains a continuum of estimates of different flavor. We enlarge the spectrum of methods by combining decoupling with quadratic Weyl sum estimates, to address new cases of interest. All results are proved in the general framework of real analytic curves.

The genus of curves in {mathbb {P}}^4 and {mathbb {P}}^5 not contained in quadrics

A classical problem in the theory of projective curves is the classification of all their possible genera in terms of the degree and the dimension of the space where they are embedded. Fixed integers r, d, s, Castelnuovo-Halphen’s theory states a sharp upper bound for the genus of a non-degenerate, reduced and irreducible curve of degree d in {mathbb {P}}^r, under the condition of being not contained in a surface of degree <s. This theory can be generalized in several ways. For instance, fixed integers r, d, k, one may ask for the maximal genus of a curve of degree d in {mathbb {P}}^r, not contained in a hypersurface of degree <k. In the present paper we examine the genus of curves C of degree d in {mathbb {P}}^r not contained in quadrics (i.e. h^0({mathbb {P}}^r, {mathcal {I}}_C(2))=0). When r=4 and r=5, and dgg 0, we exhibit a sharp upper bound for the genus. For certain values of rge 7, we are able to determine a sharp bound except for a constant term, and the argument applies also to curves not contained in cubics.

Algebraic and geometric characterizations of a class of Algebraic-Hyperbolic Pythagorean-Hodograph curves

In this paper, we study algebraic and geometric characteristics of a class of Algebraic-Hyperbolic curves that possess Pythagorean-Hodograph (PH) properties, which are called AHPH curves. These curves are defined on the space Γn=span{1,t,…,tn−3,sinh⁡t,cosh⁡t}, n∈N﹨{1,2}. We prove that all non-degenerate AHPH curves belong to Γ4. For this particular case, we give an explicit expression in terms of canonical basis functions. To reveal the geometric properties of AHPH curves, we study the geometric constraints on their Bézier like control polygons. The main result shows that an AH curve is an AHPH curve if and only if the interior angles of its control polygon are equal, and the second leg-length of the control polygon is the geometric mean of the other two leg-lengths. Our main idea is to represent a planar parametric curve in complex form. As an application, we give some examples of G1 Hermite interpolation using AHPH curves. We point out that there are no more than two AHPH curves for any given G1 Hermite conditions.

The moduli space of tropical curves with fixed Newton polygon

Abstract Given a lattice polygon, we study the moduli space of all tropical plane curves with that Newton polygon. We determine a formula for the dimension of this space in terms of combinatorial properties of that polygon. If this polygon is nonhyperelliptic or maximal and hyperelliptic, then this formula matches the dimension of the moduli space of nondegenerate algebraic curves with the given Newton polygon.

Sobolev improving for averages over curves in [formula omitted

We study Lp-Sobolev improving for averaging operators Aγ given by convolution with a compactly supported smooth density μγ on a non-degenerate curve. In particular, in 4 dimensions we show that Aγ maps Lp(R4) to the Sobolev space L1/pp(R4) for all 6<p<∞. This implies the complete optimal range of Lp-Sobolev estimates, except possibly for certain endpoint cases. The proof relies on decoupling inequalities for a family of cones which decompose the wave front set of μγ. In higher dimensions, a new non-trivial necessary condition for Lp(Rn)→L1/pp(Rn) boundedness is obtained, which motivates a conjectural range of estimates.

Diophantine approximation on curves and the distribution of rational points: Contributions to the divergence theory

In this paper we develop an explicit method for studying the distribution of rational points near manifolds. As a consequence we obtain optimal lower bounds on the number of rational points of bounded height lying at a given distance from an arbitrary non-degenerate curve in Rn. This generalises previous results for analytic non-degenerate curves. Furthermore, the main results are proved in the inhomogeneous setting. For n≥3, the inhomogeneous aspect is new even under the additional assumption of analyticity. Applications of the main distribution theorem also include the inhomogeneous Khintchine-Jarník type theorem for divergence for arbitrary non-degenerate curves in Rn.

Space curves, X-ranks and cuspidal projections

Let Xsubset mathbb {P}^3 be an integral and non-degenerate curve. We say that qin mathbb {P}^3setminus X has X-rank 3 if there is no line Lsubset mathbb {P}^3 such that qin L and #(Lcap X)ge 2. We prove that for all hyperelliptic curves of genus gge 5 there is a degree g+3 embedding Xsubset mathbb {P}^3 with exactly 2g+2 points with X-rank 3 and another embedding without points with X-rank 3 but with exactly 2g+2 points qin mathbb {P}^3 such that there is a unique pair of points of X spanning a line containing q. We also prove the non-existence of points of X-rank 3 for general curves of bidegree (a, b) in a smooth quadric (except in known exceptional cases) and we give lower bounds for the number of pairs of points of X spanning a line containing a fixed qin mathbb {P}^3setminus X. For all integers gge 5, xge 0 we prove the existence of a nodal hyperelliptic curve X with geometric genus g, exactly x nodes, deg (X) = x+g+3 and having at least x+2g+2 points of X-rank 3.

Correction to: Non-congruent non-degenerate curves with identical signatures

Correction to: Non-congruent non-degenerate curves with identical signatures

Non-congruent non-degenerate curves with identical signatures

While the equality of differential signatures (Calabi et al, Int. J. Comput. Vis. 26: 107-135, 1998) is known to be a necessary condition for congruence, it is not sufficient (Musso and Nicolodi, J. Math Imaging Vis. 35: 68-85, 2009). Hickman (J. Math Imaging Vis. 43: 206-213, 2012, Theorem 2) claimed that for non-degenerate planar curves, equality of Euclidean signatures implies congruence. We prove that while Hickman's claim holds for simple, closed curves with simple signatures, it fails for curves with non-simple signatures. In the later case, we associate a directed graph with the signature and show how various paths along the graph give rise to a family of non-congruent, non-degenerate curves with identical signatures. Using this additional structure, we formulate congruence criteria for non-degenerate, closed, simple curves and show how the paths reflect the global and local symmetries of the corresponding curve.