论文标题
罗斯定理的塔式范围有流行的差异
Tower-type bounds for Roth's theorem with popular differences
论文作者
论文摘要
绿色开发了一种算术规律性引理，以证明罗斯定理在密集集中的算术进展方面得到了加强。它指出，每$ε> 0 $都有一些$ n_0（ε）$，以便每$ n \ ge n_0（ε）$和$ a \ aubset [n] $带有$ | a | =αn$，有一些非零$ d $，因此$ a $至少包含$（α^3 - ε）n $三$三 - 算术算术进展，差异为$ d $。 我们证明，格林定理中的最低$ n_0（ε）$是$ \ log（1/ε）$的指数高度为2s的高度塔。上限和上限都是新的。它表明，在本应用中使用规律性引理产生的较高型界限是定量的。
Green developed an arithmetic regularity lemma to prove a strengthening of Roth's theorem on arithmetic progressions in dense sets. It states that for every $ε> 0$ there is some $N_0(ε)$ such that for every $N \ge N_0(ε)$ and $A \subset [N]$ with $|A| = αN$, there is some nonzero $d$ such that $A$ contains at least $(α^3 - ε) N$ three-term arithmetic progressions with common difference $d$. We prove that the minimum $N_0(ε)$ in Green's theorem is an exponential tower of 2s of height on the order of $\log(1/ε)$. Both the lower and upper bounds are new. It shows that the tower-type bounds that arise from the use of a regularity lemma in this application are quantitatively necessary.
