Harman Patil (Editor)

Vinogradov's mean value theorem

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Vinogradov's mean value theorem is an important inequality in analytic number theory, named for I. M. Vinogradov. It relates to upper bounds for J s , k ( X ) , the number of solutions to the system of k simultaneous Diophantine equations in 2 s variables given by

Contents

x 1 j + x 2 j + ⋯ + x s j = y 1 j + y 2 j + ⋯ + y s j ( 1 ≤ j ≤ k )

with

1 ≤ x i , y i ≤ X , ( 1 ≤ i ≤ s ) .

In other words, an estimate is provided for the number of equal sums of k-th powers of integers up to X. An alternative analytic expression for J s , k ( X ) is

J s , k ( X ) = ∫ [ 0 , 1 ) k | f k ( α ; X ) | 2 s d α

where

f k ( α ; X ) = ∑ 1 ≤ x ≤ X exp ⁡ ( 2 π i ( α 1 x + ⋯ + α k x k ) ) .

A strong estimate for J s , k ( X ) is an important part of the Hardy-Littlewood method for attacking Waring's problem and also for demonstrating a zero free region for the Riemann zeta-function in the critical strip. Various bounds have been produced for J s , k ( X ) , valid for different relative ranges of s and k . The classical form of the theorem applies when s is very large in terms of k .

On December 4, 2015, Jean Bourgain, Ciprian Demeter, and Larry Guth announced a proof of Vinogradov's Mean Value Theorem.

The conjectured form

By considering the X s solutions where

x i = y i , ( 1 ≤ i ≤ s )

one can see that J s , k ( X ) ≫ X s .

A more careful analysis (see Vaughan equation 7.4) provides the lower bound

J s , k ≫ X s + X 2 s − 1 2 k ( k + 1 ) .

The main conjecture of Vinogradov's mean value theorem is that the upper bound is close to this lower bound. More specifically that for any ϵ > 0 we have

J s , k ( X ) ≪ X s + ϵ + X 2 s − 1 2 k ( k + 1 ) + ϵ .

If

s ≥ 1 2 k ( k + 1 )

this is equivalent to the bound

J s , k ( X ) ≪ X 2 s − 1 2 k ( k + 1 ) + ϵ .

Similarly if s ≤ 1 2 k ( k + 1 ) the conjectural form is equivalent to the bound

J s , k ( X ) ≪ X s + ϵ .

Stronger forms of the theorem lead to an asymptotic expression for J s , k , in particular for large s relative to k the expression

J s , k ∼ C ( s , k ) X 2 s − 1 2 k ( k + 1 ) ,

where C ( s , k ) is a fixed positive number depending on at most s and k , holds.

Vinogradov's bound

Vinogradov's original theorem of 1935 showed that for fixed s , k with

s ≥ k 2 log ⁡ ( k 2 + k ) + 1 4 k 2 + 5 4 k + 1

there exists a positive constant D ( s , k ) such that

J s , k ( X ) ≤ D ( s , k ) ( log ⁡ X ) 2 s X 2 s − 1 2 k ( k + 1 ) + 1 2 .

Although this was a ground-breaking result, it falls short of the full conjectured form. Instead it demonstrates the conjectured form when

ϵ > 1 2 .

Subsequent improvements

Vinogradov's approach was improved upon by Karatsuba and Stechkin who showed that for s ≥ k there exists a positive constant D ( s , k ) such that

J s , k ( X ) ≤ D ( s , k ) X 2 s − 1 2 k ( k + 1 ) + η s , k ,

where

η s , k = 1 2 k 2 ( 1 − 1 k ) [ s k ] ≤ k 2 e − s / k 2 .

Noting that for

s > k 2 ( 2 log ⁡ k − log ⁡ ϵ )

we have

η s , k < ϵ ,

this proves that the conjectural form holds for s of this size.

The method can be sharpened further to prove the asymptotic estimate

J s , k ∼ C ( s , k ) X 2 s − 1 2 k ( k + 1 ) ,

for large s in terms of k .

In 2012 Wooley improved the range of s for which the conjectural form holds. He proved that for

k ≥ 2 and s ≥ k ( k + 1 )

and for any ϵ > 0 we have

J s , k ( X ) ≪ X 2 s − 1 2 k ( k + 1 ) + ϵ .

Ford and Wooley have shown that the conjectural form is established for small s in terms of k . Specifically they show that for

k ≥ 4

and

1 ≤ s ≤ 1 4 ( k + 1 ) 2

for any ϵ > 0

we have

J s , k ( X ) ≪ X s + ϵ .

References

Vinogradov's mean-value theorem Wikipedia
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