Waring

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Warings Problem

• M. Ram Murty, FRSC, FNA, FNASc
• Queens Research Chair
• Queens University

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Lagranges theorem

• In 1770, Lagrange proved that every natural
  number can be written as a sum of four squares.
• This was first conjectured by Bachet in 1621 who
  verified the conjecture for every number less
  than 326.

Joseph Louis Lagrange (1736-1813)
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Fermat and Euler

• Fermat claimed a proof of the four square
  theorem.
• But the first documented proof of a fundamental
  step in the proof was taken by Euler.

Pierre de Fermat (1601-1665)
Leonhard Euler (1707-1783)
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Edward Waring and Meditationes Algebraicae

• In 1770, Waring wrote in his book Meditationes
  Algebraicae that every natural number can be
  written as a sum of four squares, as a sum of
  nine cubes, as a sum of 19 fourth powers and so
  on.
• This is called Warings problem.

Edward Waring (1736-1798)
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The problem with cubes

• Can every natural number be expressed as a sum of
  9 cubes?
• This was first proved in 1908 by Arthur Wieferich
  (1884-1954)
• Wieferich was a high school teacher and wrote
  only five papers in his entire life.
• But all of these papers were of high quality.
• A small error in Wieferichs paper was corrected
  by A.J. Kempner in 1912.

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What about fourth powers?

• Theorem (Balasubramanian, Deshouillers, Dress,
  1986) Every number can be written as a sum 19
  fourth powers.

R. Balasubramanian
J.-M. Deshouillers
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So what exactly is Warings problem?

• For each natural number k, there is a number g
  g(k) such that every number can be written as a
  sum of g kth powers.
• Moreover g(2)4, g(3)9, g(4)19 and so on.
• The first question is if g(k) exists.
• The second is what is the formula (if there is
  one) for g(k)?

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Hilberts Theorem

• Theorem (Hilbert, 1909) For each k, there is a
  gg(k) such that every number can be written as a
  sum of g kth powers.

David Hilbert (1862-1943)
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What is g(k)?

• J.A. Euler (the son of L. Euler) conjectured in
  1772 that g(k) 2k (3/2)k 2.
• The number 2k(3/2)k-1 lt3k can only use 1s and
  2ks when we try to write it as a sum of kth
  powers.
• The most frugal choice is (3/2)k -1 2ks
  followed by 1s.
• This gives g(k) 2k (3/2)k 2.
• Thus, g(1)1, g(2)4, g(3)9, g(4)19.
• g(5)37 (J. Chen, 1964)
• g(6)73 (S. Pillai, 1940)

J. Chen (1933-1996)
S.S. Pillai (1901-1950)
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Pillais Theorem

• Write 3k 2kq r, with 0 lt r lt 2k.
• If rq 2k, then g(k) 2k (3/2)k 2.
• Equivalent formulation if (3/2)k1-(3/4)k,
  then g(k) 2k (3/2)k 2.
• Mahler (1957) proved that this condition holds
  for all k sufficiently large. However, his proof
  was ineffective since it uses Roths theorem in
  Diophantine approximation which is ineffective.

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The circle method

• In his letter to Hardy written in 1912, Ramanujan
  alluded to a new method called the circle method.
  

S. Ramanujan (1887-1920)
This was developed by Hardy and Littlewood in
several papers and the method is now called the
Hardy-Littlewood method.
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The function G(k)

• Define G(k) as follows. For each k there is an
  no(k) such that every n no(k) can be written as
  a sum of G(k) kth powers.
• Clearly G(k) g(k).
• Note that g(k) 2k (3/2)k 2 implies that
  g(k) has exponential growth.
• Using the circle method, Vinogradov in 1947
  showed that G(k)k(2log k 11)
• Hardy Littlewood conjectured that G(k)lt4k and
  this is still an open problem.

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Schnirelmans theorem

• Let A be an infinite set and set A(n) be the
  number of elements of A less than or equal to n.
• If B is another infinite set, then what can we
  say about the set AB ab a e A, b e B?
• Here we allow for the empty choice.
• For example, is there a relation between A(n),
  B(n), and (AB)(n)?

L. Schnirelman (1905-1938)
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Schnirelmans density

• Define the density of A as d(A) infn1 A(n)/n.
• Thus, A(n)dn for all values of n.
• Note the density of even numbers is zero
  according to this definition since A(1)0. The
  density of odd numbers is ½.
• d(A)1 if and only if A is the set all natural
  numbers.
• Theorem (Schnirelman, 1931)
• d(AB)d(A)d(B)-d(A)d(B).

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An elementary approach

• Let A(n)s, B(n)t. Write a1 lt a2 lt ... lt as n.
• Let ri B(ai1 ai -1) and write these numbers
  as b1 lt b2 lt ... lt bri
• Then ai lt ai b1 lt ai b2 lt ... lt ai brilt
  ai1
• Therefore, (AB)(n) is at least
• A(n) r1 r2 ... rs-1 B(n-as) B(a1-1)
• A(n)
• d(B)((a1-1) (a2-a1-1) ... (as-as-1-1)
  (n-as) )
• A(n) d(B)(n-s) (1-d(B))A(n)d(B)n
• (1-d(B))d(A)n d(B)n.

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An application of induction

• So we have d(AB) 1-(1-d(A))(1-d(B)).
• By induction, we have
• d(A1 ... As) 1 (1-d(A1))...(1-d(As))
• Notation 2A AA, 3A AAA, etc.
• Corollary. If d(A)gt0, then for some t, we have
  d(tA) gt ½.
• Proof. By the above, d(tA) 1-(1-d(A))t.
• If d(A)1, we are done. Suppose 0ltd(A)lt1.
• Then, (1-d(A))t tends to zero as t tends to
  infinity.

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What happens if d(A)gt1/2?

• Then A(n) gt n/2.
• This is the size of A a e A, an.
• Consider the set B n a a e A, a n.
• This set has size A(n) gt n/2.
• If B and A are disjoint, we get more than n
  numbers which are n, a contradiction.
• Thus, every number can be written as a sum of two
  elements of A.

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Consequence of Schnirelmans Theorem

• If A has positive Schnirelman density, then for
  some t, we have tA is the set of natural numbers.
• In other words, every number can be written as a
  sum of at most t elements from the set A.
• Let us apply this observation to count the number
  of numbers n which can be written as a sum of t
  kth powers.

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X1k X2k ... Xtk n

• Let A be the set of numbers that can be written
  as a sum of t kth powers.
• Key lemma The number of solutions is at most
  A(n)nt/k-1.
• On the other hand, a lower bound is given by
  (n/t)1/kt .
• This gives that A(n) has positive Schnirelman
  density.

U.V. Linnik (1915-1972)
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Some open problems

• Can every sufficiently large number be written as
  a sum of 6 cubes? (Unknown)
• The conjecture is that G(3)4. What we know is
  that 4G(3)7.
• Hypothesis K (Hardy and Littlewood) Let rg,k(n)
  be the number of ways of writing n as a sum of g
  kth powers. Then, rk,k(n) O(ne) for any egt0.
• G(k)max(k1, ?(k)) where ?(k) is the smallest
  value of g predicted by local obstructions.

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THANK
YOU!