aa r X i v : . [ m a t h . N T ] O c t A NOTE ON RATIONAL POINTS NEAR PLANAR CURVES
SAM CHOW
Abstract.
Under fairly natural assumptions, Huang counted the numberof rational points lying close to an arc of a planar curve. He obtained upperand lower bounds of the correct order of magnitude, and conjectured anasymptotic formula. In this note, we establish the conjectured asymptoticformula. Introduction
Let f be a real-valued function defined on a compact interval I = [ ρ, ξ ] ⊆ R .For positive real numbers δ / Q >
1, define˜ N f ( Q, δ ) = ( ( a, b, q ) ∈ Z : 1 q Q, a/q ∈ I, gcd( a, b, q ) = 1 , | f ( a/q ) − b/q | < δ/Q ) . Roughly speaking, this counts the number of rational points with denominatorat most Q that lie within δQ − of the curve C f = { ( x, f ( x )) : x ∈ I } . Huang[3, Theorem 2] estimated this quantity. As discussed in [3], such estimates arereadily applied to the Lebesgue theory of metric diophantine approximation. Theorem 1.1 (Huang) . Let < c c . Assume that f : I → R is a C function satisfying c | f ′′ ( x ) | c ( x ∈ I ) , with Lipschitz second derivative. Assume further that / > δ > Q ε − , (1.1) for some ε ∈ (0 , . Then √ ζ (3) + O ( Q − ε/ ) ˜ N f ( Q, δ ) | I | δQ ζ (3) + O ( Q − ε/ ) . (1.2) The implied constant depends on
I, c , c , ε and the Lipschitz constant; it isindependent of f, δ and Q . Theorem 1.1 sharpened the upper bounds obtained by Huxley [4] and Vaughan–Velani [5], as well as the lower bounds obtained by Beresnevich–Dickinson–Velani [1] and Beresnevich–Zorin [2].
Mathematics Subject Classification.
Key words and phrases.
Metric diophantine approximation, rational points near curves.
The purpose of this note is to squeeze together the constants in (1.2), so asto confirm Huang’s conjectured asymptotic formula˜ N f ( Q, δ ) ∼ ζ (3) | I | δQ ( Q → ∞ ) , (1.3)within the range (1.1). The asymptotic formula (1.3) follows straightforwardlyfrom our theorem, which we state below and establish in the next section. Theorem 1.2.
Assume the hypotheses of Theorem 1.1. Let η > and < τ < ε/ . Then ζ (3) − η + O ( Q − τ ) ˜ N f ( Q, δ ) | I | δQ ζ (3) + η + O ( Q − τ ) . The implied constant depends on
I, c , c , ε, η and the Lipschitz constant. We use Landau and Vinogradov notation: for functions f and positive-valued functions g , we write f ≪ g or f = O ( g ) if there exists a constant C such that | f ( x ) | Cg ( x ) for all x . If S is a set, we denote the cardinality of S by S .The author is supported by EPSRC Programme Grant EP/J018260/1, andthanks Faustin Adiceam for a discussion.2. The count
In this section, we prove Theorem 1.2. For positive real numbers δ / Q >
1, define the auxiliary counting functionˆ N f ( Q, δ ) = ( ( a, b, q ) ∈ Z : 1 q Q, a/q ∈ I, gcd( a, b, q ) = 1 , | f ( a/q ) − b/q | < δ/q ) . With the same assumptions as in Theorem 1.1, Huang [3, Corollary 1] showedthat ˆ N f ( Q, δ ) = ( ζ (3) − + O ( Q − ε/ )) · | I | δQ . (2.1)Let t ∈ N , 1 / < α < α i = α i (0 i t ) . We will have t ≪ η
1, so the hypothesis (1.1) is satisfied with 2 τ in place of ε and ( α i Q, α j δ ) in place of ( Q, δ ), whenever Q is large and 0 i, j t . Inparticular (2.1) holds with these adjustments, soˆ N f ( α i Q, α j δ ) = (cid:16) α i α j ζ (3) + O ( Q − τ ) (cid:17) · | I | δQ (0 i, j t ) . (2.2) NOTE ON RATIONAL POINTS NEAR PLANAR CURVES 3
Employing (2.2), we have˜ N f ( Q, δ ) > t X i =1 ( ( a, b, q ) ∈ Z : α i Q < q α i − Q, a/q ∈ I, gcd( a, b, q ) = 1 , | f ( a/q ) − b/q | < α i δ/q ) = t X i =1 ( ˆ N f ( α i − Q, α i δ ) − ˆ N f ( α i Q, α i δ ))= t X i =1 (cid:16) α i − α i − α i ζ (3) + O ( Q − τ ) (cid:17) · | I | δQ . Now ˜ N f ( Q, δ ) > (cid:16) X ( α ) ζ (3) + O ( tQ − τ ) (cid:17) · | I | δQ , (2.3)where X ( α ) = X i t ( α i − α i − α i ) . We compute that X ( α ) = ( α − α ) t − X j =0 ( α ) j = ( α − α )(1 − α t )1 − α = (1 − α t )(1 − (1 + α + α ) − ) . Choosing α close to 1, and then choosing t ≪ η X ( α ) > / − ζ (3) η. Substituting this into (2.3) yields the desired lower bound.We attack the upper bound in a similar fashion, but there is an extra termto consider. By (2.2), we have˜ N f ( Q, δ ) − ˜ N f ( α t Q, α t δ ) t X i =1 ( ( a, b, q ) ∈ Z : α i Q < q α i − Q, a/q ∈ I, gcd( a, b, q ) = 1 , | f ( a/q ) − b/q | < α i − δ/q ) = t X i =1 ( ˆ N f ( α i − Q, α i − δ ) − ˆ N f ( α i Q, α i − δ ))= t X i =1 (cid:16) α i − − α i − α i ζ (3) + O ( Q − τ ) (cid:17) · | I | δQ . Now ˜ N f ( Q, δ ) − ˜ N f ( α t Q, α t δ ) (cid:16) Y ( α ) ζ (3) + O ( tQ − τ ) (cid:17) · | I | δQ , where Y ( α ) = X i t ( α i − − α i − α i ) . SAM CHOW
Here Y ( α ) = α − X ( α ) − α − α = 1 + α α + α . Choosing α close to 1 gives Y ( α ) / ζ (3) η/
2, and so˜ N f ( Q, δ ) ˜ N f ( α t Q, α t δ ) + (cid:16) ζ (3) + η O ( tQ − τ ) (cid:17) · | I | δQ . (2.4)For the first term on the right hand side of (2.4), we bootstrap Huang’supper bound (1.2). This gives˜ N f ( α t Q, α t δ ) (cid:16) α t ζ (3) + O ( Q − τ ) (cid:17) · | I | δQ . Choosing t ≪ η α t ζ (3) η/
2, we now have˜ N f ( α t Q, α t δ ) (cid:16) η O ( Q − τ ) (cid:17) · | I | δQ . Substituting this into (2.4) provides the sought upper bound, completing theproof of the theorem.
References [1] V. Beresnevich, D. Dickinson and S. Velani,
Diophantine approximation on planarcurves and the distribution of rational points , Ann. of Math. (2) (2007), 367–426,with an Appendix II by R.C. Vaughan.[2] V. Beresnevich and E. Zorin,
Explicit bounds for rational points near planar curves andmetric Diophantine approximation , Adv. Math. (2010) 3064–3087.[3] J.-J. Huang,
Rational points near planar curves and Diophantine approximation , Adv.Math. (2015), 490–515.[4] M. N. Huxley,
The rational points close to a curve , Ann. Sc. Norm. Super. Pisa Cl. Sci.(4) (1994) 357–375.[5] R. C. Vaughan and S. Velani, Diophantine approximation on planar curves: the con-vergence theory , Invent. Math. (2006), 103–124.
Department of Mathematics, University of York, Heslington, York, YO105DD, United Kingdom
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