On simultaneous rational approximations to a real number, its square, and its cube

@inproceedings{Roy2008OnSR,
  title={On simultaneous rational approximations to a real number, its square, and its cube},
  author={Damien Roy},
  year={2008}
}
where c = c(n, ξ) > 0 is an appropriate constant depending only on n and ξ, and where τ(2) = 2, τ(3) = (3 + √ 5)/2, τ(4) = 3 and τ(n) = b(n+ 1)/2c if n ≥ 5. For n = 2, 3, this value of τ(n) cannot be improved (see [3] for the case n = 2 and [7] for the case n = 3). For n ≥ 4, M. Laurent showed in [4] that τ(n) can be taken to be d(n + 1)/2e. However, at present, no optimal value for τ(n) is known for any single value of n ≥ 4. Furthermore, we possess no non-trivial upper bound for τ(n) for n… CONTINUE READING

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