On Certain Sets of Positive Density
On Certain Sets of Positive Density
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DOI：
10.1112/jlms/s1-34.3.358
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发表时间：
1959-07
影响因子：
1.2
通讯作者：
P. Varnavides
中科院分区：
文献类型：
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作者：
P. Varnavides
其中Gd（x）表示对于固定d的良好级数的数目。实际上，每个不小于Jed或大于x-kd的α i包含在恰好满足u^ l，u-1-（k-l）d<xj的k个级数中，因为存在u的k个值使得u= c^（modd）且w< af^-wf（k-l）d。由于kd< Sx/k，这些的数量至少为8x-28 x/k= 8（1-2fk）x。用f（u，d）表示级数u，u-1-d，.中a的个数，其中求和扩展到u> 1，u-1-（k-l）d> 1，u-1-（k-l）d> 1，u-1-（< aj. The number of progressions with tt>k-l）d> 1。x明显小于x，因此我们有z）k。（五）
Gd (x)> i8x,'(3) where Gd (x) denotes the number of good progressions for a fixed d. Indeed every a {, which is not less than Jed or greater than x—kd is contained in exactly k progressions satisfying u^ l, u-\-(k—l) d< x; for there are k values of u such that u= c^(modd) and w< af^-wf (&—l) d. Since kd< Sx/k, the number of these at is at least 8x—28x/k= 8 (1—2fk) x. Denote by f (u, d) the number of a's in the progression u, u-\-d,..., uj-(k—l) d; then Y, f (u) d)> k8 (l--^\x> l8kx {k> S),(4) where the summation is extended over u> 1, u-\-{k—\) d< aj. The number of progressions with tt> 1, u-\-{k—\) d^. x is clearly less than x and therefore we have z) k.(5)