On the constant factor in Vinogradov's mean value theorem (Q2717590)

Vinogradov's Mean Value Theorem gives an upper bound for the integral NEWLINE\[NEWLINEJ_k(b,Q)=\int_0^1\dotsb \int _0^1 \left |\sum_{x=1}^Q e(\alpha _1x + \alpha _2 x^2 + \dotsb + \alpha _kx^k)\right |^{2b} d\alpha _1 \dotsb d\alpha _k.NEWLINE\]NEWLINE Lower bounds are also of theoretical interest, and the author obtains a new result in this direction, namely that \(J_k(b,Q) > (1!2!\dotsb k!)(b+1)^{-k}Q^{2b-k(k+1)/2}\). The main novelty in the proof is to interpret \(J_k(b,Q)\) as a system involving binomial coefficients \(\binom xj \) instead of powers \(x^j\).