On a theorem of Nathanson on Diophantine approximation

Abstract In 1974, M. B. Nathanson proved that every irrational number α represented by a simple continued fraction with infinitely many elements greater than or equal to k , is approximable by an infinite number of rational numbers p/q satisfying <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mi>α</m:mi> <m:mo>−</m:mo> <m:mi>p</m:mi> <m:mrow> <m:mo>/</m:mo> </m:mrow> <m:mi>q</m:mi> <m:mrow> <m:mo>|</m:mo> </m:mrow> <m:mo>&lt;</m:mo> <m:mn>1</m:mn> <m:mrow> <m:mo>/</m:mo> </m:mrow> <m:mo stretchy="false">(</m:mo> <m:msqrt> <m:msup> <m:mi>k</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>+</m:mo> <m:mn>4</m:mn> </m:msqrt> <m:msup> <m:mi>q</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> <m:mo>.</m:mo> </m:math> $|\alpha-p/q|&lt;1/(\sqrt{k^2+4}q^2).$ In this paper, we refine this result.