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Binary Quadratic Forms in Difference Sets. (arXiv:1810.03680v1 [math.NT])
来源于:arXiv
We show that if $h(x,y)=ax^2+bxy+cy^2\in \mathbb{Z}[x,y]$ satisfies $b^2\neq
4ac$, then any subset of $\{1,2,\dots,N\}$ with no nonzero differences in the
image of $h$ has size at most a constant depending on $h$ times
$N\exp(-c\sqrt{\log N})$, where $c=c(h)>0$. We achieve this goal by adapting an
$L^2$ density increment strategy previously used to establish analogous results
for sums of one or more single-variable polynomials. Our exposition is thorough
and self-contained, in order to serve as an accessible gateway for readers who
are unfamiliar with previous implementations of these techniques. 查看全文>>