Alan R. H. Baker

On a Theorem of Sprindzuk

It is proved that if Φ(q) denotes a positive monotonic decreasing function of the integer variable q < 0 such that Σq-1Φ(q) converges then for almost all real θ and any positive integer n only a finite number of integers q exist satisfying max||qθi|| < 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 (1964a, b) in which a long-standing conjecture of Mahler from the theory of transcendental numbers is established.

Studies of field systems in the British Isles

Preface 1. Introduction: Materials and methods 2. Field systems north west England 3. Field systems of Northumberland and Durham 4. Field systems of Yorkshire 5. Field systems of the West Midlands 6. Field systems of the East Midlands 7. Field systems of East Anglia 8. Field systems of the Chiltern Hills 9. Field systems of Southeast England 10. Field systems of North Wales 11. Field systems of South Wales 12. Field systems of Scotland 13. Field systems of Ireland 14. Conclusion: Problems and perspectives.