Let \(P_ x\) be the plane through the point (x,0) on the x-axis in \({\mathbb{R}}^ n\) which is orthogonal to the x-axis and let D(x) be the orthogonal projection of \(P_ x\cap D\) onto \(P_ 0\), where D is a region in \({\mathbb{R}}^ n\). D is said to be horn-shaped if it is connected, has a regular boundary, D(x)\(\subset D(x')\) for all \(x\geq x'\geq 0\) and all \(x\leq x'\leq 0\) and furthermore D(x) is locally integrable on \({\mathbb{R}}\). Set \(Z_ D(t)=trace(e^{t\Delta_ D}),\) where \(\Delta_ D\) is the Dirichlet Laplacian for D and define \(Z_ D(t,x)=trace(e^{t\Delta_{D(x)}}).\) The main result is that for a horn-shaped region D, all \(\delta >0\) and all t for which \(\int^{\infty}_{-\infty}Z_ D(t,x)<\infty\) then \[ | Z_ D(t)- (1/(4\pi t)^{1/2})\int^{\infty}_{-\infty}Z_ D(t,x)dx| \]
\[ \leq (1/\delta)\int^{\infty}_{-\infty}Z_ D(t,x)dx+(3+(\pi t)^{1/2}/\delta)(1/(4\pi t)^{n/2})\int^{\delta}_{-\delta}| D(x)| dx. \] Results of Fleckinger, Robert, Rozenbljum, Simon and Tamura on the distribution of the eigenvalues of \(\Delta_ D\) in the regions \(\{(x,y):\quad | x|^{\mu}\cdot | y| \leq 1,\quad \mu >0\}\subset {\mathbb{R}}^ 2\) can be recovered from the quoted result.