Properties and approximations of fractions associated to Ford circles extracted by inclined lines

Abstract

We study fractions associated to Ford circles which are extracted by means continuous curves. We show that the extracted fractions have similar properties to Farey sequences, like the Farey sum, and we prove that every ordered sequence that satisfies the Farey sum and has two adjacent fractions, can be extracted from Ford circles through continuous curves. This allows us to relate sequences of fractions that satisfy the Farey sum and continuous curves. We focus on the fractions $F_{1/m}$ extracted from inclined lines with positive slopes of the form $1/m$ and define jumps as the cardinality increments of these fractions with respect to $m$. We relate the expression for every jump to the prime omega function in terms of $m$ and find a cardinality formula related to the M\"obius function, which we approximate with three tractable expressions that grow in a log-linear way considering estimations of sums related to Euler's totient function or the graph of the lattice points corresponding to the fractions $p/q$.