论文标题
在算术进展的产品集上
On product sets of arithmetic progressions
论文作者
论文摘要
我们证明，任何有限算术进程的产品集的大小 \ [| \ MATHCAL A \ CDOT \ MATHCAL A | \ ge \ frac {| \ Mathcal a |^2} {（\ log | \ Mathcal a |）^{2θ+O（1）}}，\]，其中$2θ= 1-（1+ \ log \ log \ log \ log \ log 2）/（\ log 2）/（\ log 2）$是在庆祝的eRds乘数乘数乘数问题中的常数出现。这证实了大约二十年前的Elekes和Ruzsa的猜想。 如果相反，如果$ \ Mathcal {a} $放宽了，以作为正常密度为正的整数中有限算术进程的子集，我们证明\ [| \ Mathcal A \ CDOT \ CDOT \ MATHCAL A | \ ge \ frac {| \ Mathcal a |^{2}}} {（\ log | \ Mathcal a |）^{2 \ log 2-1 + o（1）}}。 \]这解决了Elekes和Ruzsa的另一种猜想的典型情况，其$ \ MATHCAL {A} $的产品集的大小为$ \ MATHCAL {A} $，其集合为size $ o（| \ Mathcal {a} |）$。 我们的界限急切到指数中的$ O（1）$项。我们进一步证明了上述结果的不对称扩展。
We prove that the size of the product set of any finite arithmetic progression $\mathcal{A}\subset \mathbb{Z}$ satisfies \[|\mathcal A \cdot \mathcal A| \ge \frac{|\mathcal A|^2}{(\log |\mathcal A|)^{2θ+o(1)} } ,\] where $2θ=1-(1+\log\log 2)/(\log 2)$ is the constant appearing in the celebrated Erdős multiplication table problem. This confirms a conjecture of Elekes and Ruzsa from about two decades ago. If instead $\mathcal{A}$ is relaxed to be a subset of a finite arithmetic progression in integers with positive constant density, we prove that \[|\mathcal A \cdot \mathcal A | \ge \frac{|\mathcal A|^{2}}{(\log |\mathcal A|)^{2\log 2- 1 + o(1)}}. \] This solves the typical case of another conjecture of Elekes and Ruzsa on the size of the product set of a set $\mathcal{A}$ whose sumset is of size $O(|\mathcal{A}|)$. Our bounds are sharp up to the $o(1)$ term in the exponents. We further prove asymmetric extensions of the above results.
