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

$x_1^j+x_2^j+\cdots+x_s^j=y_1^j+y_2^j+\cdots+y_s^j\quad (1\le j\le k)$

with

$1\le x_i,y_i\le X, (1\le i\le s)$.

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

$f_k(\mathbf\alpha;X)=\sum_{1\le x\le X}\exp(2\pi i(\alpha_1x+\cdots+\alpha_kx^k)).$

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] An expository paper by Bourgain is "On the Vinogradov mean value".

The conjectured form

By considering the $X^s$ solutions where

$x_i=y_i, (1\le i\le s)$

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

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

$J_{s,k}\gg X^s+X^{2s-\frac12k(k+1)}.$

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

$J_{s,k}(X)\ll X^{s+\epsilon}+X^{2s-\frac12k(k+1)+\epsilon}.$

If

$s\ge k(k+1)$

this is equivalent to the bound

$J_{s,k}(X)\ll X^{2s-\frac12k(k+1)+\epsilon}.$

Similarly if $s\le k(k+1)$ 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

$J_{s,k}\sim \mathcal C(s,k)X^{2s-\frac12k(k+1)},$

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 ^[4] showed that for fixed $s,k$ with

$s\ge k^2\log (k^2+k)+\frac14k^2+\frac54 k+1$

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

$J_{s,k}(X)\le D(s,k)(\log X)^{2s}X^{2s-\frac12k(k+1)+\frac12}.$

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

$\epsilon>\frac12$.

Subsequent improvements

Vinogradov's approach was improved upon by Karatsuba ^[5] and Stechkin ^[6] who showed that for $s\ge k$ there exists a positive constant $D(s,k)$ such that

$J_{s,k}(X)\le D(s,k)X^{2s-\frac12k(k+1)+\eta_{s,k}},$

where

$\eta_{s,k}=\frac12 k^2\left(1-\frac1k\right)^{\left[\frac sk\right]}\le k^2e^{-s/k^2}.$

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

$J_{s,k}\sim \mathcal C(s,k)X^{2s-\frac12k(k+1)},$

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

In 2012 Wooley ^[7] improved the range of $s$ for which the conjectural form holds. He proved that for

$k\ge 2$ and $s\ge k(k+1)$

and for any $\epsilon>0$ we have

$J_{s,k}(X)\ll X^{2s-\frac12k(k+1)+\epsilon}.$

Ford and Wooley^[8] have shown that the conjectural form is established for small $s$ in terms of $k$. Specifically they show that for

$k\ge 4$

and

$1\le s\le \frac14(k+1)^2$

for any $\epsilon>0$

we have

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

References

1. E. C. Titchmarsh (rev. D. R. Heath-Brown): The theory of the Riemann Zeta-function, OUP
2. Bourgain, J.; Demeter, Ciprian; Guth, Larry (2015). "Proof of the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three". arXiv:1512.01565.
3. R.C. Vaughan: The Hardy-Littlewood method, CUP
4. I. M. Vinogradov, New estimates for Weyl sums, Dokl. Akad. Nauk SSSR 8 (1935), 195–198
5. A. A. Karatsuba, The mean value of the modulus of a trigonometric sum, Izv. Akad. Nauk SSSR 37 (1973), 1203–1227.
6. S. B. Stechkin, On mean values of the modulus of a trigonometric sum, Trudy Mat. Inst. Steklov 134 (1975), 283–309.
7. T. D. Wooley, Vinogradov’s mean value theorem via efficient congruencing, Annals of Math. 175 (2012), 1575–1627.
8. Kevin Ford and Trevor D. Wooley: On Vinogradov’s mean value theorem: strong diagonal behaviour via efficient congruencing http://www.math.uiuc.edu/~ford/wwwpapers/ec3vindiag.pdf