If now we have two irrational numbers α and β, can we find sequences { a n }, { b n } and { q n } with q n → ∞ as n → ∞ such that | α − a n q n | &lt; 1 q n 2 and | β − b n q n | &lt; 1 q n 2 ?

Question

If now we have two irrational numbers   α and   β, can we find sequences   {      a    n    },   {      b    n    } and   {      q    n    } with       q    n    →  ∞ as   n  →  ∞ such that      |    α  −            a      n              q      n            |    &amp;lt;      1          q      n      2         and       |    β  −            b      n              q      n            |    &amp;lt;      1          q      n      2        ?

Samuel Hooper · Accepted Answer

The simultaneous version of Dirichlet&#039;s theorem asserts that you can find infinitely many       a    n    ,      b    n   and       q    n   such that      |    α    −                  a        n                    q        n              |    &amp;lt;      1          q      n              3                  /                2                     and           |    β    −                  b        n                    q        n              |    &amp;lt;      1          q      n              3                  /                2              .For almost all pairs of real numbers (including, via a theorem of Schmidt, pairs of independent algebraic numbers), the exponent   3      /    2 is best possible.

If now we have two irrational numbers alpha and beta, can we find sequences {a_n}, {b_n} and {q_n} with q_n->oo as n->oo such that |alpha−a_n/q_n|<1/q_n^2 and |beta−b_n/q_n|<1/q_n^2?

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