Diophantine inequalities for polynomial rings
The author studies the Hardy-Littlewood method for the Laurent series field \(\mathbb F_q ((1/T))\) over the finite field \(\mathbb F_q\) with \(q\) elements. He shows that if \(\lambda_1\), \(\lambda_2\), \(\lambda_3\) are nonzero elements in \(\mathbb F_q ((1/T))\) satisfying \(\lambda_1/\lambda_2\not\in \mathbb F_q(T)\) and \(\text{sgn} (\lambda_1)+ \text{sgn} (\lambda_2)+ \text{sgn} (\lambda_3)= 0\), then the values of the sum \[ \lambda_1P_1+ \lambda_2P_2+ \lambda_3P_3, \] as \(P_i\) \((i=1,2,3)\) run independently through all monic irreducible polynomials in \(\mathbb F_q[T]\), are everywhere dense on the ``non-Archimedean'' line \(\mathbb F_q ((1/T))\), where \(\text{sgn}(f)\in\mathbb F_q\) denotes the leading coefficient of \(f\in \mathbb F_q ((1/T))\). A more explicit inequality is given in the paper. In the proof, the integral \[ \int_{\mathbb K_\infty} E(af) \chi_n(a) \,da= \begin{cases} 1&\text{ if }\deg f< n,\\ 0 &\text{ if }\deg f\geq n, \end{cases}, \] plays a role entirely analogous to the integral invoked in the Hardy-Littlewood method on the real line, where \(E\) is an additive character and \(\chi\) the characteristic function of the maximal ideal of \(\mathbb F_q (1/T)\).
scientific article; zbMATH DE number 1347929