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On the mean lattice point discrepancy of a convex disc. (English) Zbl 1013.11065

Let \(D\) be a convex planar domain \((x(s),y(s))\) with 4 times continuously differentiable functions \(x(s)\) and \(y(s)\) satisfying \(0<C_1\leq |x''(s) y'(s) -x'(s)y''(s)|\leq C_2\) for some fixed constants \(C_1\), \(C_2\). Define the lattice point discrepancy \[ P_D(t)\equiv \#(tD\cap Z^2)-\text{area}(D)t^2. \] M. N. Huxley proved that \(P_D(t)\ll t^{46/73}(\log t)^{315/146}\). It is conjectured that \(\inf\{\theta\in\mathbb{R}: P_D(t)\ll t^\theta\} =1/2\). W. G. Nowak proved earlier that \(\int^T_0(P_D(t))^2\, dt\ll T^2\) which supports the conjecture. P. Bleher improved this by proving the asymptotic formula \(\int^T_0(P_D(t))^2\, dt\sim C(D)T^2\). M. N. Huxley obtained a short-interval result \(\int^{T+1/2}_{T-1/2}(P_D(t))^2\, dt\ll T\log T.\)
In this paper the author sharpens this result by proving that for any \(c>0\), \[ \int^{T+c\log T}_{T-c\log T} (P_D(t))^2\, dt\ll T\log T. \]

MSC:

11P21 Lattice points in specified regions
11K38 Irregularities of distribution, discrepancy
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