Vinogradov's mean-value theorem

By considering the ${\displaystyle X^{s}}$ solutions where

${\displaystyle x_{i}=y_{i},(1\leq i\leq s)}$

one can see that ${\displaystyle J_{s,k}(X)\gg X^{s}}$.

A more careful analysis (see Vaughan [3] equation 7.4) provides the lower bound

${\displaystyle J_{s,k}\gg X^{s}+X^{2s-{\frac {1}{2}}k(k+1)}.}$

The main conjecture of Vinogradov's mean value theorem was that the upper bound is close to this lower bound. More specifically that for any ${\displaystyle \epsilon >0}$ we have

${\displaystyle J_{s,k}(X)\ll X^{s+\epsilon }+X^{2s-{\frac {1}{2}}k(k+1)+\epsilon }.}$

This was proved by Jean Bourgain, Ciprian Demeter, and Larry Guth[4] and by a different method by Trevor Wooley.[5]

If

${\displaystyle s\geq {\frac {1}{2}}k(k+1)}$

this is equivalent to the bound

${\displaystyle J_{s,k}(X)\ll X^{2s-{\frac {1}{2}}k(k+1)+\epsilon }.}$

Similarly if ${\displaystyle s\leq {\frac {1}{2}}k(k+1)}$ the conjectural form is equivalent to the bound

${\displaystyle J_{s,k}(X)\ll X^{s+\epsilon }.}$

Stronger forms of the theorem lead to an asymptotic expression for ${\displaystyle J_{s,k}}$, in particular for large ${\displaystyle s}$ relative to ${\displaystyle k}$ the expression

${\displaystyle J_{s,k}\sim {\mathcal {C}}(s,k)X^{2s-{\frac {1}{2}}k(k+1)},}$

where ${\displaystyle {\mathcal {C}}(s,k)}$ is a fixed positive number depending on at most ${\displaystyle s}$ and ${\displaystyle k}$, holds, see Theorem 1.2 in.[6]

Vinogradov's original theorem of 1935 [7] showed that for fixed ${\displaystyle s,k}$ with

${\displaystyle s\geq k^{2}\log(k^{2}+k)+{\frac {1}{4}}k^{2}+{\frac {5}{4}}k+1}$

there exists a positive constant ${\displaystyle D(s,k)}$ such that

${\displaystyle J_{s,k}(X)\leq D(s,k)(\log X)^{2s}X^{2s-{\frac {1}{2}}k(k+1)+{\frac {1}{2}}}.}$

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

${\displaystyle \epsilon >{\frac {1}{2}}}$.

Vinogradov's approach was improved upon by Karatsuba[8] and Stechkin[9] who showed that for ${\displaystyle s\geq k}$ there exists a positive constant ${\displaystyle D(s,k)}$ such that

${\displaystyle J_{s,k}(X)\leq D(s,k)X^{2s-{\frac {1}{2}}k(k+1)+\eta _{s,k}},}$

where

${\displaystyle \eta _{s,k}={\frac {1}{2}}k^{2}\left(1-{\frac {1}{k}}\right)^{\left[{\frac {s}{k}}\right]}\leq k^{2}e^{-s/k^{2}}.}$

Noting that for

${\displaystyle s>k^{2}(2\log k-\log \epsilon )}$

we have

${\displaystyle \eta _{s,k}<\epsilon }$,

this proves that the conjectural form holds for ${\displaystyle s}$ of this size.

The method can be sharpened further to prove the asymptotic estimate

${\displaystyle J_{s,k}\sim {\mathcal {C}}(s,k)X^{2s-{\frac {1}{2}}k(k+1)},}$

for large ${\displaystyle s}$ in terms of ${\displaystyle k}$.

In 2012 Wooley[10] improved the range of ${\displaystyle s}$ for which the conjectural form holds. He proved that for

${\displaystyle k\geq 2}$ and ${\displaystyle s\geq k(k+1)}$

and for any ${\displaystyle \epsilon >0}$ we have

${\displaystyle J_{s,k}(X)\ll X^{2s-{\frac {1}{2}}k(k+1)+\epsilon }.}$

Ford and Wooley[11] have shown that the conjectural form is established for small ${\displaystyle s}$ in terms of ${\displaystyle k}$. Specifically they show that for

${\displaystyle k\geq 4}$

and

${\displaystyle 1\leq s\leq {\frac {1}{4}}(k+1)^{2}}$

for any ${\displaystyle \epsilon >0}$

we have

${\displaystyle J_{s,k}(X)\ll X^{s+\epsilon }.}$