tailieunhanh - Analytic Number Theory A Tribute to Gauss and Dirichlet Part 9

Tham khảo tài liệu 'analytic number theory a tribute to gauss and dirichlet part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 152 BEN GREEN Definition . Fix an integer k 3. We define rk N to be the largest cardinality of a subset A c 1 . N which does not contain k distinct elements in arithmetic progression. Erdos and Turán asked simply what is rk N To this day our knowledge on this question is very unsatisfactory and in particular we do not know the answer to Question . Is it true that rk N n N for N N0 k If this is so then the primes contain k-term arithmetic progressions on density grounds alone irrespective of any additional structure that they might have. I do not know of anyone who seriously doubts the truth of this conjecture and indeed all known lower bounds for rk N are much smaller than n N . The most famous such bound is Behrend s assertion Beh46 that rs N . slightly superior lower bounds are known for rk N k 4 cf. LL Ran61 . The question of Erdos and Turan became and remains rather notorious for its difficulty. It soon became clear that even seemingly modest bounds should be regarded as great achievements in combinatorics. The first really substantial advance was made by Klaus Roth who proved Theorem Roth Rot53 . We have r3 N c N loglogN -1. The key feature of this bound is that log log N tends to infinity with N albeit slowly2. This means that if one fixes some small positive real number such as and then takes a set A c 1 . N containing at least integers then provided N is sufficiently large this set A will contain three distinct elements in arithmetic progression. The generalisation of this statement to general k remained unproven until Sze-meredi clarified the issue in 1969 for k 4 and then in 1975 for general k. His result is one of the most celebrated in combinatorics. Theorem Szemeredi Sze69 Sze75 . We have rk N o N for any fixed k 3. Szemeráedi s theorem is one of many in this branch of combinatorics for which the bounds if they are ever worked out are almost unimaginably weak. Although it is in principle possible to obtain an explicit .