[HN Gopher] Simple math moves the needle
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       Simple math moves the needle
        
       Author : digital55
       Score  : 49 points
       Date   : 2023-09-29 15:05 UTC (2 days ago)
        
 (HTM) web link (www.quantamagazine.org)
 (TXT) w3m dump (www.quantamagazine.org)
        
       | fiforpg wrote:
       | Kakeya problem is a beautiful question, and Besicovitch's
       | construction has been really useful in harmonic analysis
       | (Fefferman's _The multiplier problem for the ball_ is a classic
       | example). Still, in the eyes of wider public it must appear
       | uncanny, requiring the Amazon truck analogies, etc.
       | 
       | Reminds me of Vladimir Arnold's comment on the state of modern
       | number theory (presumably talking about density of primes in
       | arithmetic progressions, he says): why would you even _want_ to
       | add primes, they were born to be multiplied?
        
         | gumby wrote:
         | > why would you even want to add primes, they were born to be
         | multiplied?
         | 
         | I actually laughed out loud at this. For real! So great.
        
       | ducttapecrown wrote:
       | Why is the Kakeya problem so popular?
        
         | fiforpg wrote:
         | If I understand correctly, the initial appeal goes back to the
         | early days of measure theory.
         | 
         | Say, a 2-dimensional set is cut parallel to y-axis, and the
         | cuts all have length 1. If the x-coordinates of the cuts
         | themselves have length 1, you know that the 2-dimensional set
         | must be "large". This is because you can integrate the cut
         | length, and tell that it has area 1. (Think of a 1x1 square for
         | an illustration).
         | 
         | In Kakeya's problem, the cuts still have length 1, but are no
         | longer parallel to any one axis. Besicovitch's construction
         | shows, Kakeya set can have very small area, yet contain cuts of
         | length 1 in many directions. This situation is quite different.
         | 
         | This counterexample turned out to be useful in other areas of
         | pure math, some discussed in the wiki entry for Kakeya problem.
        
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       (page generated 2023-10-01 23:01 UTC)