Fourier uniformity on subspaces
Fourier uniformity on subspaces
复制标题
子空间上的傅立叶均匀性
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
T. Sanders
中科院分区：
文献类型：
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作者：
B. Green;T. Sanders
Let $mathbb{F}$ be a fixed finite field, and let $A subset mathbb{F}^n$. It is a well-known fact that there is a subspace $V leq mathbb{F}^n$, $mbox{codim} V ll_{delta} 1$, and an $x$, such that $A$ is $delta$-uniform when restricted to $x + V$ (that is, all non-trivial Fourier coefficients of $A$ restricted to $x + V$ have magnitude at most $delta$). We show that if $mathbb{F} = mathbb{F}_2$ then it is possible to take $x = 0$; that is, $A$ is $delta$-uniform on a subspace $V leq mathbb{F}^n$. We give an example to show that this is not necessarily possible when $mathbb{F} = mathbb{F}_3$.
ADDED July 2016: shortly after this paper appeared on the arxiv, F. Manners showed us a rather short argument he had found in 2013, giving a better bound for our main theorem. We do not, therefore, intend to publish this note. The example over $mathbb{F}_3$ may still be of interest to some readers and so we will not withdraw the paper from the arxiv.
