ROTH’S THEOREM FOR FOUR VARIABLES AND ADDITIVE STRUCTURES IN SUMS OF SPARSE SETS
ROTH’S THEOREM FOR FOUR VARIABLES AND ADDITIVE STRUCTURES IN SUMS OF SPARSE SETS
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稀疏集和中四变量和可加结构的罗斯定理
DOI：
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发表时间：
2014
期刊：
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通讯作者：
Olof Sisask
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文献类型：
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作者：
T. Schoen;Olof Sisask
本文证明了：如果A子集{1，ldots，N}$不包含方程$x+y+z= 3 w $的任何非平凡解，则$[001 pdf 1st-31files]|一|leqslant frac{N}{exp（c（log N）^{1/7}）}，end{eqnarray}$$其中$c>0$是某个绝对常数。根据贝伦德的构造，这个界是正确的：指数1/7不能被任何大于1/2的常数所取代。我们还建立了一个相关的结果，它说如果A $subset {1，ldots，N}$，和集A+A+A$包含长算术级数，或者如果A $subset mathbb{F}_{q}^{n}$，和集A + A + A $包含高维仿射子空间，即使A$具有上述形状的密度。
We show that if $Asubset {1,ldots ,N}$ does not contain any nontrivial solutions to the equation $x+y+z=3w$ , then $$egin{eqnarray}|A|leqslant frac{N}{exp (c(log N)^{1/7})},end{eqnarray}$$ where $c>0$ is some absolute constant. In view of Behrend’s construction, this bound is of the right shape: the exponent $1/7$ cannot be replaced by any constant larger than $1/2$ . We also establish a related result, which says that sumsets $A+A+A$ contain long arithmetic progressions if $Asubset {1,ldots ,N}$ , or high-dimensional affine subspaces if $Asubset mathbb{F}_{q}^{n}$ , even if $A$ has density of the shape above.