On the number of lattice points in certain planar segments

Rok, strany: 2003, 173 - 187

Let $D₀\subset {\Bbb R}²$ be a compact domain whose boundary is a simple closed curve composed of finitely many pieces such that on each piece the radius of curvature exists everywhere, is bounded and non-zero, and is continuously differentiable with respect to the tangent angle. Further, let $D$ be a plane domain obtained by applying a rigid motion to $D₀$ and let $D(a,b):=\{(x,y)\in D: y≥ ax+b\}$, where $a,b\in\Bbb R$. Generalizing Huxley's famous theorem we show that when $a$ is taken from a large class $R$ of irrational numbers and $b$ is arbitrary, for a real parameter $λ$

$$ \#(λD\cap\Bbb Z²) =λ² area D +O(λ^0.63)    ( λ\to∞ ) . $$

Thereby the $O$@-constant depends only on the basic domain $D₀$ and the class $R$. Additionally, we are able to extend the applicability of the standard method of estimating rounding error sums of the shape

$$ Ψ(f;u,v;λ):=\kern -1mm ∑\limits_{uλ≤ n≤ vλ}\kern -2mm ψ (λ f (\frac nl ))    ( λ\to∞ ) , $$

where $ψ(z)=z-[z]-1/2$ and $f$ is a real-valued function defined on an interval $[u,v]\subset\Bbb R$ with continuous derivatives up to order $3$ and the property that $f''$ does not vanish on $[u,v]$. By Huxley's method, $Ψ(f;u,v;λ) \llλ^0.63$ under the additional condition that $f'''$ does not vanish on $[u,v]$. We show that this condition, which has always been interpreted as technical, is superfluous.

ISO 690:

Kuba, G. 2003. On the number of lattice points in certain planar segments. In Mathematica Slovaca, vol. 53, no.2, pp. 173-187. 0139-9918.

APA:

Kuba, G. (2003). On the number of lattice points in certain planar segments. Mathematica Slovaca, 53(2), 173-187. 0139-9918.