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 December 8th, 2015, 04:37 PM #1 Newbie   Joined: Dec 2015 From: israel Posts: 2 Thanks: 0 Probabilistic methods and equations over $m$-dimensional space. Given a set $A$ of $n$ different points in the space $(\mathbb{Z}_p)^m$ (assume $p$ is prime), and given $\delta>0$. show the following property holds for a big enough $n$ and $p$ (you can demand that $n$ is dependent on $p$; you can't demand anything from $m$ except that it's big enough so our space can actually contain $n$ different points): show that there exists a set of linear equations (that look like "$u\bullet v=x$") such that the number of points that don't solve any equation is in the interval $(\frac{1}{2} \pm \delta)n$. there should be $O(p \log n)$ equations. I came across this question why studying probabilistic methods and algorithms, specifically in a chapter on the second moment method. [Please excuse the the bad translation, the questions isn't originally in English. Last edited by ShadiEndrawis; December 8th, 2015 at 04:59 PM. December 11th, 2015, 08:56 AM   #2
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 Originally Posted by ShadiEndrawis Given a set $A$ of $n$ different points in the space $(\mathbb{Z}_p)^m$ (assume $p$ is prime), and given $\delta>0$. show the following property holds for a big enough $n$ and $p$ (you can demand that $n$ is dependent on $p$; you can't demand anything from $m$ except that it's big enough so our space can actually contain $n$ different points): show that there exists a set of linear equations (that look like "$u\bullet v=x$") such that the number of points that don't solve any equation is in the interval $(\frac{1}{2} \pm \delta)n$. there should be $O(p \log n)$ equations. I came across this question why studying probabilistic methods and algorithms, specifically in a chapter on the second moment method. [Please excuse the the bad translation, the questions isn't originally in English.
I followed this thread (and your other one) but it seems that none have an answer for. May I ask what text it's from? (Perhaps something not available in English).

I am currently studying combinatorics from Jukna and am familiar (but not great at) probabilistic methods. Sorry to say I do not have an answer for you but I'm curious to learn more about the problem.

-Dave K Tags $m$dimensional, equations, methods, probabilistic, space Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post ShadiEndrawis Advanced Statistics 0 December 8th, 2015 04:35 PM Kukkurloom Geometry 7 January 14th, 2015 12:51 PM edunwaigwe@yahoo.com Economics 1 December 26th, 2012 04:57 AM OriaG Calculus 2 September 4th, 2012 12:23 PM elim Abstract Algebra 4 April 29th, 2010 10:52 AM

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