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Register a new userOn the number of rich lines in truly high dimensional sets
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Added by
Sivakanth Gopi
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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We prove a new upper bound on the number of $r$-rich lines (lines with at least $r$ points) in a `truly $d$-dimensional configuration of points $v_1,ldots,v_n in mathbb{C}^d$. More formally, we show that, if the number of $r$-rich lines is significantly larger than $n^2/r^d$ then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor $r^d$ can be replaced with a tight $r^{d+1}$. If true, this would generalize the classic Szemeredi-Trotter theorem which gives a bound of $n^2/r^3$ on the number of $r$-rich lines in a planar configuration. This conjecture was shown to hold in $mathbb{R}^3$ in the seminal work of Guth and Katz cite{GK10} and was also recently proved over $mathbb{R}^4$ (under some additional restrictions) cite{SS14}. For the special case of arithmetic progressions ($r$ collinear points that are evenly distanced) we give a bound that is tight up to low order terms, showing that a $d$-dimensional grid achieves the largest number of $r$-term progressions. The main ingredient in the proof is a new method to find a low degree polynomial that vanishes on many of the rich lines. Unlike previous applications of the polynomial method, we do not find this polynomial by interpolation. The starting observation is that the degree $r-2$ Veronese embedding takes $r$-collinear points to $r$ linearly dependent images. Hence, each collinear $r$-tuple of points, gives us a dependent $r$-tuple of images. We then use the design-matrix method of cite{BDWY12} to convert these local linear dependencies into a global one, showing that all the images lie in a hyperplane. This then translates into a low degree polynomial vanishing on the original set.
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