Errata to: the limiting curve of Jarnik's polygons

TL;DR

This paper clarifies that the limit shape of Jarnik's polygonal curve is a circle, correcting previous claims of a parabola-based shape, and references the correct established proof.

Contribution

It corrects the previously stated limit shape of Jarnik's polygons, confirming the circle as the true limit shape with proper references.

Findings

The limit shape of Jarnik's polygons is a circle.

Previous claims of a parabola-based shape are incorrect.

The correct result has been established in prior work.

Abstract

In this note we point out that the limit shape of the Jarnk polygonal curve [1] is the circle, not a curve consisting of arcs of parabolas as it has been stated in the main result of [2] (Section 2). The correct result has been established and proven in [4].

Equations

$$V_t=\left\{(q,a)\in {\mathbf{Z}}^2:\gcd (q,a)=1;\sqrt{q^2+a^2}\le t\right\}.$$

$$V_Q=\left\{(q,a)\in {\mathbf{Z}}^2:\gcd (q,a)=1;\max \{|a|,|q|\}\le Q\right\}.$$

Full text

• In this note we point out that the limit shape of Jarník’s polygonal curve [1] is the circle, not a curve consisting of arcs of parabolas as it has been stated in the main result of [2] (Section 2). The correct result has been established and proven in [4].

1 Introduction

In order to establish the maximum number $f(s)$ of lattice points on a strictly convex curve of length at most $s$, Jarník [1] has considered a family of convex lattice polygons. Vertices of these polygons lie on strictly convex curves which optimize $f(s)$, in an asymptotic sense. It has been shown [4] that there is a limit shape of such polygons. More precisely, as the number of vertices of these polygons increase and after a proper scaling (e.g., for a factor equal to the perimeter of the related polygon) these polygons converge to a circle. The result has been derived starting from the family of the convex lattice polygons whose edge slopes $q/a$ belong to the set $V_{t}$, defined as follows (for an arbitrary $t>1$):

[TABLE]

Another family of convex lattice polygons has been considered in [2]. The edge slopes $q/a$ of lattice polygons from this family make the set $V_{Q},$ defined for all $Q>1$, as follows

[TABLE]

The polygons $V_{Q}$ do not relate to the Jarník’s polygons (i.e., do not optimize the number of the polygon vertices with respect to the Euclidean perimeter of the polygon), as it has been stated in [2]. Indeed, the observation of $V_{Q}$, as given in [2], does not lead to a conclusion about the number of lattice points on a strictly convex curve with respect to the Euclidean length of this curve.

Thus, the statement, from [2], that the limiting curve of Jarník’s polygons consist of parabolic arcs is not true. The limit shape of such polygons is a circle, as it has been shown in [4]. The result from [2] can be considered in sense of the maximal number of lattice points, on a strictly convex curve, with respect to the curve perimeter taken in sense of $l_{\infty}$ distance, rather than in sense of the $l_{2}$ (i.e. Euclidean) distance.

It is also worth mentioning the following: The some of the results from [2] (see Section 2) are derived following the idea and technique from [4] and [5], but this was not referenced properly; The term ‘limit shape’ has been used by the others (see [3], for an example) but in different sense than it has been done in [2], [4], and [5].