AI 中文总结
研究前\(n\)个平方数中\(B_2[g]\)子集的最大基数，用\(2(g + 1)\) - 均匀超图方法，证明其至少为一仅依赖于\(g\)的常数乘\(n^{\frac{2g}{2g + 1}}(\log n)^{\frac{2 - 2^g}{2g + 1}}\)。
AI 中文摘要
我们证明，对于每个固定整数\(g\geq1\)，前\(n\)个平方数的\(B_2[g]\)子集中的最大基数，对于所有足够大的\(n\)，至少是一个仅依赖于\(g\)的正常数乘以\(n^{\frac{2g}{2g + 1}}(\log n)^{\frac{2 - 2^g}{2g + 1}}\)。对于\(g = 1\)，这恢复了Lefmann和Thiele关于前\(n\)个平方数的Sidon子集的定理。证明遵循他们的超图方法，但用编码\(g + 1\)个表示为两个平方数之和的\(2(g + 1)\) - 均匀超图代替编码两个这样表示的\(4\) - 均匀超图。要点是验证这个更高均匀性的超图边少且\(2\) - 圈少；然后根据Duke、Lefmann和Rödl关于不拥挤超图的独立定理得出下界。
英文摘要
We prove that, for every fixed integer $g\geq 1$, the largest cardinality of a $B_2[g]$ subset of the first $n$ squares is at least a positive constant, depending only on $g$, times $$ n^{\frac{2g}{2g+1}}(\log n)^{\frac{2-2^g}{2g+1}}, $$ for all sufficiently large $n$. For $g=1$, this recovers the theorem of Lefmann and Thiele on Sidon subsets of the first squares. The proof follows their hypergraph method, but replaces the $4$-uniform hypergraph encoding two representations as a sum of two squares by a $2(g+1)$-uniform hypergraph encoding $g+1$ such representations. The main point is to verify that this higher-uniformity hypergraph has few edges and few $2$-cycles; the lower bound then follows from an independence theorem for uncrowded hypergraphs due to Duke, Lefmann and Rödl.
