Show that an irrational number ξ = [ a 0 ; a 1 , ⋯ ] has bounded partial quotients a k if and only if there exists δ &gt; 0 such that | ξ − p q | &gt; δ q 2 for all rational p q .

Question

Show that an irrational number   ξ  =  [      a    0    ;      a    1    ,  ⋯  ] has bounded partial quotients       a    k   if and only if there exists   δ  &amp;gt;  0 such that      |    ξ    −          p      q        |    &amp;gt;      δ          q      2      for all rational       p    q  .

Samuel Vang · Accepted Answer

Take   δ  &amp;lt;  1      /    2. If       |    ξ    −          p      q        |    &amp;lt;      δ          q      2      then   p      /    q  =      p    k        /        q    k   is a convergent of the continued fraction expansion of   ξ and      1                  q        k            (              q        k            +              q                  k          +          1                    )        &amp;lt;      |    ξ    −                  p        k                    q        k              |    &amp;lt;      1                  q        k                    q                  k          +          1                      .But       a          k      +      1        q  &amp;lt;      q    2    &amp;lt;  (      a          k      +      1        +  1  )      q    2   since       q          k      +      1        =      a          k      +      1            q    k    +      q          k      −      1        .. If       a    k    &amp;lt;  M for some   M then      |    ξ    −          p      q        |    &amp;gt;      δ          q      2      for   δ  =      1          M      +      1        .. This proves one direction.Conversely, suppose      |    ξ    −          p      q        |    &amp;gt;      δ          q      2        .Then       a    k    &amp;lt;  1      /        δ    ,, so that       a    k   is bounded.

Show that an irrational number ξ = [ a 0 </msub> ;

Answered question

Answer & Explanation

New Questions in Algebra I