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I'm trying to understand how to use the circle method to derive an asymptotic formula for Waring's Problem. Do so using the circle method developed by Hardy and Littlewood. In doing this, I want to make sure that:

given $P, Q$ natural numbers with $P > 1$ and $Q \ge 2P$, the following intervals don't overlap:

$$\{ c: |c- (a/q)| \le 1/(pQ) \}$$ where

  • $q \le P$
  • $1\le a \le q $
  • $(a,q) = 1$

Sorry, I can't seem to get Latex commands right. Any insight to this would be helpful, thanks!

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thanks Andrew for the edits! –  user45793 Oct 23 '12 at 22:34

1 Answer 1

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Here is an extended hint: let $I(a,q),I(a',q')$ be two such intervals, and suppose that $I(a,q)\cap I(a',q')\neq \varnothing.$ Use this to show that $\lvert aq'-a'q\rvert\le \dfrac{q+q'}{Q}.$ If we suppose that $q\neq q',$ then the left hand side cannot be zero (why?). On the other hand, in this case we can see that $q+q'<2P,$ which will show that the left hand side must be less than one, yielding a contradicion. So we must have $q=q'.$ This case follows by a similar method.

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thanks for the suggestion Andrew. I understand the method for the proof, but I don't see how the first inequality could hold though. If its true then the rest of the proof follows, but I can't rationalize it holding. –  user45793 Oct 23 '12 at 23:19
    
Dear @Zak, the first inequality follows by applying the triangle inequality to $|c-\frac{a}{q}|+|c-\frac{a'}{q'}|$ where $c\in I(a,q)\cap I(a',q').$ (I.e., assuming the intersection is nonempty.) –  Andrew Oct 23 '12 at 23:22
    
you're absolutely right. thanks for the help! this solves a large block I was having –  user45793 Oct 23 '12 at 23:24
    
You're welcome! –  Andrew Oct 23 '12 at 23:26

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