Integer Points near Smooth Plane Curves

In what follows, N1$N \geqslant 1$ is an arbitrary large integer, δ(0,12)$\delta \in \left( 0,\frac{1}{2} \right)$ and if f:[N,2N]R$f : [N,2N] \longrightarrow \mathbb {R}$ is any positive function, then let R(f,N,δ)$\mathcal {R}(f,N,\delta)$ be the number of integers n[N,2N]$n \in [N,2N]$ such that f(n)<δ$\lVert f(n) \rVert < \delta$, where as usual x$\lVert x \rVert$ denotes the distance from xR$x \in \mathbb {R}$ to its nearest integer. Note that, since δ$\delta$ is very small, R(f,N,δ)$\mathcal {R}(f,N,\delta)$ roughly counts the number of integer points very close to the arc y=f(x)$y = f(x)$ with Nx2N$N \leqslant x \leqslant 2N$. Hence the trivial estimate is given by R(f,N,δ)N+1$\mathcal {R}(f,N,\delta) \leqslant N+1$. The number R(f,N,δ)$\mathcal {R}(f,N,\delta)$ arises fairly naturally in a large collection of problems in number theory, e.g. [Filaseta, 1990 †Filaseta, M. 1990
Short interval results for squarefree numbers
J. Number Theory, 35, 128--149.
], [Filaseta & Trifonov, 1996 †Filaseta, M., & Trifonov, O. 1996
The distribution of fractional parts with applications to gap results in number theory
Proc. Lond. Math. Soc., III. Ser., 73(2), 241--278.
], [Huxley, 1996 †Huxley, M.N. 1996
Area, Lattice Points and Exponential Sums
Oxford Science Pub.
], [Huxley & Sargos, 1995 †Huxley, M.N., & Sargos, P. 1995
Integer points close to a plane curve of class Cn$C^n$. (Points entiers au voisinage d'une courbe plane de classe Cn$C^n$.)
Acta Arith., 69(4), 359--366.
], [Huxley & Sargos, 2006 †Huxley, M.N., & Sargos, P. 2006
Integer points in the neighborhood of a plane curve of class Cn$C^n$. II. (Points entiers au voisinage d'une courbe plane de classe Cn$C^n$. II.)
Funct. Approximatio, Comment. Math., 35, 91--115.
], [Huxley & Trifonov, 1996 †Huxley, M.N., & Trifonov, O. 1996
The square-full numbers in an interval
Math. Proc. Camb. Phil. Soc., 119, 201--208.
] and . We deal with either getting an asymptotic formula of the shape
R(f,N,δ)=Nδ+Error terms
where the remainder terms depend on the derivatives of f$f$ but not on δ$\delta$, or finding an upper bound for R(f,N,δ)$\mathcal {R}(f,N,\delta)$ as accurate as possible.
1. Bounds using elementary methods
The basic result of the theory is well-known and may be found in [Vinogradov, 2004 †Vinogradov, I.M. 2004
The method of trigonometrical sums in the theory of numbers
Mineola, NY: Dover Publications Inc. Translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport, Reprint of the 1954 translation.
]. The proof follows from a clever use of the mean-value theorem (see Theorem~5.6 of [Bordellès, 2012 †Bordellès, O. 2012
Arithmetic Tales
Universitext. Springer London Heidelberg New York Dordrecht.
] for instance).
Theorem (First derivative test)
Let fC1[N,2N]$f \in C^1 [N,2N]$ such that there exist λ1>0$\lambda_1 >0$ and c11$c_1 \geqslant 1$ such that, for all x[N,2N]$x \in [N,2N]$, we have
λ1f(x)c1λ1.
Then
R(f,N,δ)2c1Nλ1+4c1Nδ+2δλ1+1.
This result is useful when λ1$\lambda_1$ is very small, so that the condition is in general too restrictive in the applications. Using a rather neat combinatorial trick, [Huxley, 1996 †Huxley, M.N. 1996
Area, Lattice Points and Exponential Sums
Oxford Science Pub.
] succeeded in passing from the first derivative to the second derivative. This reduction step enables him to apply this theorem to a function being approximatively of the same order of magnitude as f$f'$. This provides the following useful result.
Theorem (Second derivative test)
Let fC2[N,2N]$f \in C^2[N,2N]$ such that there exist λ2>0$\lambda_2 >0$ and c21$c_2 \geqslant 1$ such that, for all x[N,2N]$x \in [N,2N]$, we have
λ2f′′(x)c2λ2andNλ2c12.
Then
R(f,N,δ)6{(3c2)1/3Nλ1/32+(12c2)1/2Nδ1/2+1}.
Both hypotheses above are often satisfied in practice, so that this result may be considered as the first useful tool of the theory. A proof of this Theorem may be found in Theorem 5.8 of [Bordellès, 2012 †Bordellès, O. 2012
Arithmetic Tales
Universitext. Springer London Heidelberg New York Dordrecht.
]. Using a k$k$th version of Huxley's reduction principle may allow us to generalize the above results. A better way is to split the integer points into two classes, namely the major arcs in which the points belong to a same algebraic curve of degree <k$< k$, and the minor arcs. The points coming from the minor arcs are treated by divided differences techniques, generalizing the proof of both theorems above and, by a careful analysis of the points belonging to major arcs, [Huxley & Sargos, 1995 †Huxley, M.N., & Sargos, P. 1995
Integer points close to a plane curve of class Cn$C^n$. (Points entiers au voisinage d'une courbe plane de classe Cn$C^n$.)
Acta Arith., 69(4), 359--366.
] and [Huxley & Sargos, 2006 †Huxley, M.N., & Sargos, P. 2006
Integer points in the neighborhood of a plane curve of class Cn$C^n$. II. (Points entiers au voisinage d'une courbe plane de classe Cn$C^n$. II.)
Funct. Approximatio, Comment. Math., 35, 91--115.
] succeeded in proving the following fundamental result. A proof of an explicit version may be found in Theorem 5.12 of [Bordellès, 2012 †Bordellès, O. 2012
Arithmetic Tales
Universitext. Springer London Heidelberg New York Dordrecht.
].
Theorem (k$k$th derivative test)
Let k3$k \geqslant 3$ be an integer and fCk[N,2N]$f \in C^k [N,2N]$ such that there exist λk>0$\lambda_k >0$ and ck1$c_k \geqslant 1$ such that, for all x[N,2N]$x \in [N,2N]$, we have
λkf(k)(x)ckλk.
Let δ(0,14)$\delta \in \left( 0,\frac{1}{4} \right)$. Then
R(f,N,δ)αkNλ2k(k+1)k+βkNδ2k(k1)+8k3(δλk)1/k+2k2(5e3+1)
where
αk=2k2c2k(k+1)kandβk=4k2(5e3c2k(k1)k+1).
2. Bounds using exponential sums techniques
The next result leads us to estimate R(f,N,δ)$\mathcal {R}(f,N,\delta)$ with the help of exponential sums (see [Graham & Kolesnik, 1991 †Graham, S. W., & Kolesnik, G. 1991
Van der Corput's Method of Exponential Sums
London Math. Soc. Lect. Note, no. 126. Cambridge University Press.
] for instance), which have been extensively studied in the 20th century by many specialists, such as van der Corput, Weyl or Vinogradov. Nevertheless, even using the finest exponent pairs to date, the result generally does not significantly improve on the previous estimates seen above. A simple proof of the following inequality may be found in [Filaseta, 1990 †Filaseta, M. 1990
Short interval results for squarefree numbers
J. Number Theory, 35, 128--149.
].
Theorem (k$k$th derivative test)
Let f:[N,2N]R$f : [N,2N] \longrightarrow \mathbb {R}$ be any function and δ(0,14)$\delta \in \left( 0,\frac{1}{4} \right)$. Set K=(8δ)1+1$K = \left \lfloor (8 \delta)^{-1} \right \rfloor +1$. Then, for any positive integer HK$H \leqslant K$, we have
R(f,N,δ)4NH+4Hh=1HNn2Ne(hf(n)).
3. Integer points on curves
This last part is somewhat out of the scope of the TME-EMT project, but may help the reader in orienting him/herself in the litterature.

When δ0$\delta \longrightarrow 0$, we are led to counting the number of integer points lying on curves, and we denote this number by R(f,N,0)$\mathcal {R}(f,N,0)$. This problem goes back to Jarník [Jarník, 1925 †Jarník, V. 1925
Über die Gitterpunkte auf konvexen Kurven
Math. Z., 24, 500--518.
] who proved that a strictly convex arc y=f(x)$y=f(x)$ with length L$L$ has at most
3(2π)1/3L2/3+O(L1/3)
integer points and this is a nearly best possible result under the sole hypothesis of convexity. However, [Swinnerton-Dyer, 1974 †Swinnerton-Dyer, H.P.F. 1974
The number of lattice points on a convex curve
J. Number Theory, 6, 128--135.
] proved that if fC3[0,N]$f \in C^3[0,N]$ is such that |f(x)|N$|f(x)| \leqslant N$ and f′′′(x)0$f'''(x) \neq 0$ for all x[0,N]$x \in [0,N]$, then the number of integer points on the arc y=f(x)$y=f(x)$ with 0xN$0 \leqslant x \leqslant N$ is N3/5+ε$\ll N^{3/5+\varepsilon}$. This result was later generalized by [Bombieri & Pila, 1989 †Bombieri, E., & Pila, J. 1989
The number of integral points on arcs and ovals
Duke Math. J., 59, 337--357.
] who showed among other things the following estimate.
Theorem (1989)
Let N1$N \geqslant 1$, k4$k \geqslant 4$ be integers and define K=(k+22)$K = \binom{k+2}{2}$. Let I$\mathcal{I}$ be an interval with length N$N$ and fCK(I)$f \in C^K (\mathcal{I})$ satisfying |f(x)|1$|f'(x)| \leqslant 1$, f′′(x)>0$f''(x) >0$ and such that the number of solutions of the equation f(K)(x)=0$f^{(K)} (x) = 0$ is m$\leqslant m$. Then there exists a constant c0=c0(k)>0$c_0 = c_0(k) >0$ such that
R(f,N,0)c0(m+1)N1/2+3/(k+3).

Last updated on July 23rd, 2012, by Olivier Bordellès