Primitive lattice points in a thin strip along the boundary of a large convex planar domain (Q2759137)

Let \(D\) be a convex planar domain containing the origin whose boundary is in \(\mathbb C^4\) and has finite nonvanishing curvature throughout. A point \((m,n)\) in \(\mathbb Z^2\) is called primitive if \({\text gcd} (m,n)=1\). Define \(F(u)= \text{inf} \{ \tau > 0: \underline u / \tau \in D \}\) \((\underline u \in \mathfrak R^2)\) and NEWLINE\[NEWLINE\begin{aligned} \mathbf B_D (x) = &|\{\underline m = (\mathbf m_1 , \mathbf m_2) \in \mathbb Z^{\ast 2}:\mathbb F^2 (\underline m) < x, \text{gcd} (\mathbf m_1 , \mathbf m_2)=1 \}|+ \\ \tfrac 1 2 &|\{\underline m = (\mathbf m_1 , \mathbf m_2) \in \mathbb Z^{\ast 2}:\mathbb F^2 (\underline m) = x, \text{gcd} (\mathbf m_1 , \mathbf m_2)=1\}|, \end{aligned} NEWLINE\]NEWLINE where \(\mathbb Z^{\ast 2}:= \mathbb Z^2 \backslash (0,0)\}\). An asymptotical formula for \(\mathbf B_D (x)\) in the sharpest published form was obtained by Huxley and Nowak: NEWLINE\[NEWLINE \mathbf B_D(x)=6/\pi^2 \text{area} (D)x+ O (\chi^{1/2} \exp (-c(\log x)^{3/5} (\log \log x)^{-1/5})). NEWLINE\]NEWLINE The aim of this paper is to find as small as possible \(h\) for which NEWLINE\[NEWLINE \mathbf B_D(x+h)-\mathbf B_D(x)\sim 6/\pi^2 \text{area} (D)h. NEWLINE\]NEWLINE The authors prove that the above formula holds for \(h \geq \chi^{11/29}\lambda(x)\log x\) with any \(\lambda(x)\) tending to \(\infty\) with \(x\). This result is obviously better than the result which follows directly from the above Huxley-Nowak formula as well as the formula for \(\mathbf B_D(x)\) obtained by W. Miller under RH.