Sidon集在剩余類中的分布.pdf

1、南京師范大學(xué)碩士學(xué)位論文Sidon集在剩余類中的分布姓名：魏蔚申請學(xué)位級別：碩士專業(yè)：數(shù)學(xué)；基礎(chǔ)數(shù)學(xué)指導(dǎo)教師：陳永高2009-04-17南京師范大學(xué)碩士學(xué)位論文AbstractA set A of positive integers is called a Sidon set if all the sums a+b (a, b ∈A , a ≤ b) are distinct.Assume that A ? [1, n] is a Si

2、don set and |A | = (1 + o(1))n1/2. In 1997,Bernt Lindstr¨ om proved by a combinatorial method that A can be uniformlydistributed in residue classes when n is large enough. Then qualitative estimateson how the unifor

3、m distribution is are provided by Mihail N. Kolountzakis in1998.We continue this topic and give some explicit quantitative results on thedistribution in residue classes of Sidon sets. The main results are as follows:Let

4、m, n be two positive integers, A ? [1, n] be a Sidon set with r = |A |,and ri = |{a ∈ A : a ≡ i( mod m)}| (0 ≤ i < m).(1) If r ≥ n1/2, thenm?1 ?i=0?ri ? rm?2 ≤ 12 max?r4/3, r3/2m1/2?.(2) If r < n1/2, thenm?1 ?i=0?r

5、i ? rm?2 ≤ 19 max?r4/3, r3/2m1/2, r2/3(n1/2 ? r)4/3m2/3 , r(n1/2 ? r)m?.In particular, as a corollary, we obtain the following result:(3) If m ≥ n1/6 and r ≥ n1/2 ? n1/3, then for each i (0 ≤ i < m), we haveri ≤ 8n1/3