<p>Let <i>d</i> ≥ 2 be a natural number. We show that <Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2717_Article_Equa.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="360" /> </MediaObject> <EquationSource Format="TEX">\(|A-A|\geq\left(2d-2+{1 \over {d-1}}\right)|A|-(2{d^{2}}-4d+3)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo stretchy="false">∣</mo> </mrow> <mi>A</mi> <mo>−</mo> <mi>A</mi> <mrow> <mo stretchy="false">∣</mo> </mrow> <mo>≥</mo> <mrow> <mo>(</mo> <mn>2</mn> <mi>d</mi> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo stretchy="false">∣</mo> </mrow> <mi>A</mi> <mrow> <mo stretchy="false">∣</mo> </mrow> <mo>−</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mrow> <msup> <mi>d</mi> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mi>d</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </math></EquationSource> </Equation> for any sufficiently large finite subset <i>A</i> of ℝ<sup><i>d</i></sup> that is not contained in a translate of a hyperplane. By a construction of Stanchescu, this is best possible and thus resolves an old question first raised by Uhrin.</p>
Let d ≥ 2 be a natural number. We show that \(|A-A|\geq\left(2d-2+{1 \over {d-1}}\right)|A|-(2{d^{2}}-4d+3)\) for any sufficiently large finite subset A of ℝd that is not contained in a translate of a hyperplane. By a construction of Stanchescu, this is best possible and thus resolves an old question first raised by Uhrin.
