THE DUFFIN-SCHAEFFER-TYPE CONJECTURES IN VARIOUS LOCAL FIELDS

In this paper we study the Duffin-Schaeffer conjecture, which claims that lambda(boolean AND(infinity)(m=1) boolean OR(infinity)(n=m) epsilon(n)) = 1 if and only if Sigma(infinity)(n=1) lambda (epsilon(n)) = infinity, where lambda denotes the Lebesgue measure on R/Z, epsilon(n) = epsilon(n) (psi) = boolean OR(n)(m=1(m,n)=1) (m - psi(n)/n, m + psi(n)/n), and psi denotes any non-negative arithmetical function. Instead of studying the superior limit boolean AND(infinity)(m=1) boolean OR(infinity)(n=m) epsilon(n) we focus on the union boolean OR(infinity)(n=1) epsilon(n) and conjecture that there exists a universal constant C > 0 such that lambda(boolean OR(infinity)(n=1) epsilon(n)) >= C min {Sigma(infinity)(n=1) lambda(epsilon(n)), 1}. It is shown that this conjecture is equivalent to the Duffin-Schaeffer conjecture. Similar phenomena exist in the fields of p-adic numbers and formal Laurent series. Furthermore, two conjectures of Haynes, Pollington and Velani are shown to be equivalent to the Duffin-Schaeffer conjecture, and a weighted version of the second Borel-Cantelli lemma is introduced to study the Duffin-Schaeffer conjecture.