that this effective result cannot be used in practice. Upper bounds for Diophantine approximations General upper bound The first important result about upper bounds for Diophantine approximations is Dirichlet's approximation theorem, which implies that, for every irrational number , there are infinitely many fractions such that This implies immediately that one cannot suppress the in the statement of Thue-Siegel-Roth theorem. Adolf Hurwitz (1891) strengthened this result, proving that for every irrational number , there are infinitely many fractions such that Therefore, is an upper bound for the Diophantine approximations of any irrational number. The constant in this result may not be further improved without excluding some irrational numbers (see below). Émile Borel (1903) showed that, in fact, given any irrational number , and given three consecutive convergents of , at least one must satisfy the inequality given in Hurwitz's Theorem. Equivalent real numbers Definition: Two real numbers are called equivalent if there are integers with such that: So equivalence is defined by an integer Möbius transformation on the real numbers, or by a member of the Modular group , the set of invertible 2 × 2 matrices over the integers. Each rational number is equivalent to 0; thus the rational numbers are an equivalence class for this relation. The equivalence may be read on the regular continued fraction representation, as shown by the following theorem of Serret:Theorem''': Two irrational numbers x and y are equivalent if and only if there exist two positive integers h and k such that the regular continued fraction representations of x and ysatisfy for every non negative integer i. Thus, except for a finite initial sequence, equivalent numbers have the same continued fraction representation. Equivalent numbers are approximable to the same degree, in the sense that they have the same Markov constant. Lagrange spectrum As said above, the constant in Borel's theorem may not be improved, as shown by Adolf Hurwitz in 1891. Let be the golden ratio. Then for any real constant c with there are only a finite number of rational numbers such that Hence an improvement can only be achieved, if the numbers which are equivalent to are excluded. More precisely: For every irrational number , which is not equivalent to , there are infinite many fractions such that By successive exclusions — next one must exclude the numbers equivalent to — of more and more classes of equivalence, the lower bound