Exceptional sets for Diophantine inequalities (Q2929674)

Let \(k\) be a positive integer, \(0< \tau \leq 1\), and \(\mu\) a real number. Let \(\lambda_1,\ldots,\lambda_s\) be nonzero real numbers, not all in rational ratio, not all of the same sign when \(k\) is even, and at least one of which is \(\geq 2\). The authors investigate the set of exceptional \(\mu\) for which the Diophantine inequality \(|\lambda_1 x_1^k+\cdots + \lambda_s x_s^k - \mu| < \tau\) has no integral solution \((x_1,\ldots,x_s)\).NEWLINENEWLINESpecifically, let \(\mathcal{Z}_{s,k}(N,M)\) denote the set of \(\mu\in [N,N+M]\) for which the Diophantine inequality in question has no integral solutions \((x_1,\ldots,x_s)\), and let \(Z_{s,k}(N,M)\) denote the measure of this exceptional set \(\mathcal{Z}_{s,k}(N,M)\). Suppose that \(k\geq 3\), and that \((s_0,\sigma_0)\) forms a ``smooth accessible pair'' for \(k\) (defined on page 3921 of the paper under review). Let \(s\) and \(t\) be nonnegative integers with \(s\geq \max\{2k+3,17,s_0/2\}\). Then whenever \(M\geq N^{(1-1/k)^t}\), there exists a number \(\Delta >0\) such that \(Z_{s+t,k}(N,M) \ll M N^{-\Delta-(2s-s_0)\sigma_0/k}\). Therefore, taking \(M=N\) and \(t=0\) for example, the percentage of exceptional \(\mu\) in the interval \([N,2N]\) tends to zero as \(N \to \infty\).NEWLINENEWLINEThe authors provide a refined version of this bound with explicit values of \((s_0,\sigma_0)\) for \(4\leq k\leq 20\), and separately for \(k=3\) where additional control may be exercised. For larger values of \(k\) one has \(s_0=s_0(k) \sim k\log k\) and \(\sigma_0 =\sigma_0(k) \sim 1/(k\log k)\) as \(k\to \infty\). In comparison, results of this type were described when \(s\geq 2^k+1\) using the Davenport-Heilbronn method. So the machinery developed by the authors enables handling much smaller values of \(s\) with power-savings.NEWLINENEWLINEThe authors next consider the counting function \(\mathcal{N}^{\tau}_{s,k}(P;\boldsymbol{\lambda}, \mu)\) for the number of integral solutions of the Diophantine inequality \(|\lambda_1 x_1^k+\cdots + \lambda_s x_s^k - \mu| < \tau\) in the box \(1\leq x_1,\ldots,x_s\leq P\). On isolating the main term \(2\tau \Omega_{s,k}(\boldsymbol{\lambda},\mu P^{-k}) P^{s-k}\) via a heuristic application of the Davenport-Heilbronn method, and taking \(P=N^{1/k}\), the measure \(\tilde{Z}_{s,k}^{\tau}(N;\psi;\boldsymbol{\lambda})\) of the set of numbers \(\mu \in (N/2, N]\) for which \(|\mathcal{N}^{\tau}_{s,k}(N^{1/k};\boldsymbol{\lambda}, \mu)- 2\tau \Omega_{s,k}(\boldsymbol{\lambda},\mu/N) N^{s/k-1}|> N^{s/k-1} \psi(N)^{-1}\) is considered. Here, \(\psi(N)\) is a positive function growing sufficiently slowly with \(N\) (and possibly with other parameters). In Theorems 1.4, 1.5, 1.6, and 1.7, the authors show that for suitable \(\psi\) the measure \(\tilde{Z}_{s,k}^{\tau}(N;\psi;\boldsymbol{\lambda})\) is small compared to \(N\), and they produce power-savings.NEWLINENEWLINEThe quantitative results of the authors are applicable for much smaller values of \(s\) than was previously possible. The large improvement is due to the second author's efficient congruencing approach to Vinogradov's mean value theorem. When \(s\) is somewhat smaller than is required to bound \(\tilde{Z}_{s,k}^{\tau}(N;\psi;\boldsymbol{\lambda})\) successfully, the authors obtain a nontrivial lower bound for the measure of the set of \(\mu\in [-N,N]\) for which the Diophantine inequality in question does have a solution. They also consider a Diophantine inequality over the primes; namely, the approximation of a real number \(\mu\) by a suitable linear combination in two primes \(p_1\) and \(p_2\). Suppose \(\lambda_1\) and \(\lambda_2\) are real numbers such that \(\lambda_1/\lambda_2\) is irrational. It is shown that the set of \(\mu \in [0,X]\) for which the inequality \(|\lambda_1 p_1+\lambda_2 p_2-\mu|<\tau\) has no solutions in the primes \(p_1\) and \(p_2\) is of measure \(o(X)\). So such \(\mu\) must be rare.