Abstract
It is proved that if Φ(
q
) denotes a positive monotonic decreasing function of the integer variable
q
< 0 such that Σ
q
-1
Φ(
q
) converges then for almost all real θ and any positive integer
n
only a finite number of integers
q
exist satisfying max||
q
θ
i
|| < q
-1/
n
(Φ(
q
))
n
for
j
= 1, 2, ...,
n
. Analogues are given for complex θ. The results here represent improvements on the recent work of Sprindžuk (1964
a, b
) in which a long-standing conjecture of Mahler from the theory of transcendental numbers is established.
Reference2 articles.
1. C assels J . W . S. 1951 Som e m etrical th eorem s in D io p h a n tin e ap p roxim ation . V . On a co n - jectu re o f M ahler. P roc. C am b. P h il.
2. C assels J . W . S. 1957 P ress. A n introduction to D io ph an tin e a p p ro x im a tio n . C am bridge U n IV ersity
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