http://openproblemgarden.org/op/sum_product_problem Comments for "Sum-product problem" en http://openproblemgarden.org/op/sum_product_problem <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/erdos">Erdos</a>; <a href="/category/szemeredi_endre">Szemeredi</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/number_theory_0">Number Theory</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>For every <img class="teximage" src="/files/tex/73cf4fe93b8ae2de0b0fe36a793c19bbb59a682b.png" alt="$ A \subseteq {\mathbb R} $" /> we let <img class="teximage" src="/files/tex/2941cf854a5e22795fc5c0a1750bff309068cdcf.png" alt="$ A + A = \{ a+b : a,b \in A\} $" /> and <img class="teximage" src="/files/tex/c9d6e4f1d0e09414024765ed7fb23f6f7075c83c.png" alt="$ A \cdot A = \{ ab : a,b \in A \} $" />. </p> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; For every <img class="teximage" src="/files/tex/9347bd4719a266c10e0c73946c70533f2b4266a2.png" alt="$ \epsilon &gt; 0 $" />, every sufficiently large finite set <img class="teximage" src="/files/tex/73cf4fe93b8ae2de0b0fe36a793c19bbb59a682b.png" alt="$ A \subseteq {\mathbb R} $" /> satisfies <img class="teximage" src="/files/tex/de862bc1660fe79e0bc0067ee21dd3805bbbad82.png" alt="\[ \max \{ |A \cdot A|, |A + A| \} \ge |A|^{2 - \epsilon} \]" /> </div> </tr></td> </table> </td> </tr> </table> Erdos, Paul Szemeredi, Endre product set sumset Number Theory http://openproblemgarden.org/op/sum_product_problem#comment Fri, 08 Aug 2008 00:43:21 +0200 mdevos 1821 at http://openproblemgarden.org