高阶凸性与迭代二阶矩估计
Higher Convexity and Iterated Second Moment Estimates
摘要：
证明了充分凸或近凸实数集上$N$ -元集上$$a_1 + \dots + a_k = a_1' + \dots + a_k'$$的解的个数的界。一个近凸集将是一个具有小加法加倍的集合在一个具有足够多严格单调导数的凸函数下的像。我们表明，粗略地说，每次在方程中的条款的数量增加了一倍，一个额外的节省$1$的平凡界$N^{2k-1}$的指数，从平凡的情况下$k=1$。在近凸集的上下文中，我们还提供了明确的依赖性的加法加倍参数。更高的凸性是保持这种界限所必需的，正如连续整数的完美幂集所证明的那样。我们利用这些更强的假设，使用Garaev的想法，而不是无处不在的Szemerédi-Trotter定理，这在早期的结果中还没有适应，以拥抱更高的凸性。作为一个应用，我们证明了小的改进的最佳已知的凸集的和集的额外的凸性假设下的边界。
英文摘要：
We prove bounds for the number of solutions to $$a_1 + \dots + a_k = a_1' + \dots + a_k'$$ over $N$-element sets of reals, which are sufficiently convex or near-convex. A near-convex set will be the image of a set with small additive doubling under a convex function with sufficiently many strictly monotone derivatives. We show, roughly, that every time the number of terms in the equation is doubled, an additional saving of $1$ in the exponent of the trivial bound $N^{2k-1}$ is made, starting from the trivial case $k=1$. In the context of near-convex sets we also provide explicit dependencies on the additive doubling parameters. Higher convexity is necessary for such bounds to hold, as evinced by sets of perfect powers of consecutive integers. We exploit these stronger assumptions using an idea of Garaev, rather than the ubiquitous Szemerédi-Trotter theorem, which has not been adapted in earlier results to embrace higher convexity. As an application we prove small improvements for the best known bounds for sumsets of convex sets under additional convexity assumptions.
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