Vinogradov's mean-value theorem

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 $2s$ variables given by

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

1\leq x_{i},y_{i}\leq X,(1\leq i\leq 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)=\int _{[0,1)^{k}}|f_{k}(\mathbf {\alpha } ;X)|^{2s}d\mathbf {\alpha }

where

f_{k}(\mathbf {\alpha } ;X)=\sum _{1\leq x\leq X}\exp(2\pi i(\alpha _{1}x+\cdots +\alpha _{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.^[1] 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.^[2]^[3]

The conjectured form

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

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

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

A more careful analysis (see Vaughan ^[4] equation 7.4) provides the lower bound

J_{s,k}\gg X^{s}+X^{2s-{\frac {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 $\epsilon >0$ we have

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

s\geq k(k+1)

this is equivalent to the bound

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

Similarly if $s\leq k(k+1)$ the conjectural form is equivalent to the bound

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

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

where ${\mathcal {C}}(s,k)$ is a fixed positive number depending on at most $s$ and $k$ , holds.

Vinogradov's bound

Vinogradov's original theorem of 1935 ^[5] showed that for fixed $s,k$ with

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

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

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

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

Subsequent improvements

Vinogradov's approach was improved upon by Karatsuba^[6] and Stechkin^[7] who showed that for $s\geq k$ there exists a positive constant $D(s,k)$ such that

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

where

\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

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

we have

\eta _{s,k}<\epsilon

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

for large $s$ in terms of $k$ .

In 2012 Wooley^[8] improved the range of $s$ for which the conjectural form holds. He proved that for

k\geq 2

and

s\geq k(k+1)

and for any $\epsilon >0$ we have

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

Ford and Wooley^[9] have shown that the conjectural form is established for small $s$ in terms of $k$ . Specifically they show that for

k\geq 4

and

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

for any $\epsilon >0$

we have

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

References

↑ 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 0-19-853369-1. MR 0882550.
↑ 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.
↑ Bourgain, Jean (2016-01-26). "On the Vinogradov mean value". arXiv:1512.01565.
↑ Vaughan, Robert C. (1997). The Hardy-Littlewood method. Cambridge Tracts in Mathematics. 25 (Second ed.). Cambridge: Cambridge University Press. ISBN 0-521-57347-5. MR 1435742.
↑ I. M. Vinogradov, New estimates for Weyl sums, Dokl. Akad. Nauk SSSR 8 (1935), 195–198
↑ 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.
↑ Stečkin, Sergeĭ Borisovich (1975). "Mean values of the modulus of a trigonometric sum". Trudy Mat. Inst. Steklov (in Russian). 134: 283–309. MR 0396431.
↑ Wooley, Trevor D. (2012). "Vinogradov's mean value theorem via efficient congruencing". Ann. of Math. 175 (3): 1575–1627. doi:10.4007/annals.2012.175.3.12. MR 2912712.
↑ Ford, Kevin; Wooley, Trevor D. (2014). "On Vinogradov's mean value theorem: strong diagonal behaviour via efficient congruencing". Acta Math. 213 (2): 199–236. doi:10.1007/s11511-014-0119-0. MR 3286035.

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