Vinogradov's mean-value theorem

In mathematics, Vinogradov's mean value theorem is an estimate for the number of equal sums of powers. It is an important inequality in analytic number theory, named for I. M. Vinogradov.

More specifically, let ${\displaystyle J_{s,k}(X)}$ count the number of solutions to the system of ${\displaystyle k}$ simultaneous Diophantine equations in ${\displaystyle 2s}$ variables given by

${\displaystyle x_{1}^{j}+x_{2}^{j}+\cdots +x_{s}^{j}=y_{1}^{j}+y_{2}^{j}+\cdots +y_{s}^{j}\quad (1\leq j\leq k)}$

with

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

That is, it counts the number of equal sums of powers with equal numbers of terms (${\displaystyle s}$) and equal exponents (${\displaystyle j}$), up to ${\displaystyle k}$th powers and up to powers of ${\displaystyle X}$. An alternative analytic expression for ${\displaystyle J_{s,k}(X)}$ is

${\displaystyle J_{s,k}(X)=\int _{[0,1)^{k}}|f_{k}(\mathbf {\alpha } ;X)|^{2s}d\mathbf {\alpha } }$

where

${\displaystyle f_{k}(\mathbf {\alpha } ;X)=\sum _{1\leq x\leq X}\exp(2\pi i(\alpha _{1}x+\cdots +\alpha _{k}x^{k})).}$

Vinogradov's mean-value theorem gives an upper bound on the value of ${\displaystyle J_{s,k}(X)}$.

A strong estimate for ${\displaystyle J_{s,k}(X)}$ 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.[1] Various bounds have been produced for ${\displaystyle J_{s,k}(X)}$, valid for different relative ranges of ${\displaystyle s}$ and ${\displaystyle k}$. The classical form of the theorem applies when ${\displaystyle s}$ is very large in terms of ${\displaystyle k}$.

An analysis of the proofs of the Vinogradov mean-value conjecture can be found in the Bourbaki Séminaire talk by Lillian Pierce.[2]

Lower bounds

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

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

Main conjecture and proof announcement

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 ${\displaystyle \epsilon >0}$ we have

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

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.

On December 4, 2015, Jean Bourgain, Ciprian Demeter, and Larry Guth announced a proof of Vinogradov's Mean Value Theorem.[4][5]

Vinogradov's bound

Vinogradov's original theorem of 1935 [6] 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

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

Subsequent improvements

Vinogradov's approach was improved upon by Karatsuba[7] and Stechkin[8] 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[9] 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[10] 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

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

References

1. Titchmarsh, Edward Charles (1986). The theory of the Riemann Zeta-function. Edited and with a preface by D. R. Heath-Brown (Second ed.). New York: The Clarendon Press, Oxford University Press. ISBN 978-0-19-853369-6. MR 0882550.
2. Pierce, Lilian B. (2017). "The Vinogradov mean-value theorem [after Wooley, and Bourgain, Demeter and Guth]". Séminaire Bourbaki. 69 (1134): 1–80. arXiv:1707.00119.
3. Vaughan, Robert C. (1997). The Hardy-Littlewood method. Cambridge Tracts in Mathematics. 25 (Second ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-57347-4. MR 1435742.
4. Bourgain, Jean; Demeter, Ciprian; Guth, Larry (2016). "Proof of the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three". Ann. of Math. 184 (2): 633–682. arXiv:1512.01565. doi:10.4007/annals.2016.184.2.7. hdl:1721.1/115568.
5. Bourgain, Jean (2016-01-29). "On the Vinogradov mean value". arXiv:1601.08173 [math.NT].
6. I. M. Vinogradov, New estimates for Weyl sums, Dokl. Akad. Nauk SSSR 8 (1935), 195–198
7. Karatsuba, Anatoly (1973). "Mean value of the modulus of a trigonometric sum". Izv. Akad. Nauk SSSR Ser. Mat. (in Russian). 37: 1203–1227. MR 0337817.
8. Stečkin, Sergeĭ Borisovich (1975). "Mean values of the modulus of a trigonometric sum". Trudy Mat. Inst. Steklov (in Russian). 134: 283–309. MR 0396431.
9. Wooley, Trevor D. (2012). "Vinogradov's mean value theorem via efficient congruencing". Ann. of Math. 175 (3): 1575–1627. arXiv:1101.0574. doi:10.4007/annals.2012.175.3.12. MR 2912712.
10. Ford, Kevin; Wooley, Trevor D. (2014). "On Vinogradov's mean value theorem: strong diagonal behaviour via efficient congruencing". Acta Math. 213 (2): 199–236. arXiv:1304.6917. doi:10.1007/s11511-014-0119-0. MR 3286035.