Diophantine approximation with primes and powers of two
Let \(\lambda_1\), \(\lambda_2\), \(\mu_1\), \(\mu_2\) be nonzero real numbers such that \(\lambda_1/\lambda_2\) is negative and irrational, and \(\lambda_1/\mu_1\) and \(\lambda_2/\mu_2\) are rational. Let \(\eta> 0\) be given. Then it is shown that there exists \(s_0\geq 2\) depending only on \(\lambda_1\), \(\lambda_2\), \(\mu_1\), \(\mu_2\) and \(\eta\) as follows. For any \(s> s_0\) and any real numbers \(\gamma\) and \(\mu_3,\mu_4,\dots, \mu_s\) the inequality \[ |\lambda_1 p_1+ \lambda_2 p_2+ \mu_1 2^{n_1}+\cdots+ \mu_2 2^{n_s}+ \gamma|< \eta \] has infinitely many solutions. This can be seen as a Diophantine approximation analogue of \textit{Yu. V. Linnik}'s result [Mat. Sb., N. Ser. 32(74), 3--60 (1953; Zbl 0051.03402)] on representations of integers using 2 primes and a bounded number of powers of 2. The proof combines the Davenport-Heilbronn circle method with a recent approach to Linnik's theorem, due to the reviewer and \textit{J.-C. Puchta} [Asian J. Math. 6, 535--566 (2002; Zbl 1097.11050)].
scientific article; zbMATH DE number 2028913