Let A ■ ■_(N),and f_(A)(s)={1-|A/N,-|A|/N,for s ∈A,for s■ A.We define the pseudorandom measure of order k of the subset A as follows,P _(k)(A,N)=max D|∑n∈■_(N)|f_(A)(n+c_(1))f_(A)(n+c_(2))…f_(A)(n+c_(k))|where...Let A ■ ■_(N),and f_(A)(s)={1-|A/N,-|A|/N,for s ∈A,for s■ A.We define the pseudorandom measure of order k of the subset A as follows,P _(k)(A,N)=max D|∑n∈■_(N)|f_(A)(n+c_(1))f_(A)(n+c_(2))…f_(A)(n+c_(k))|where the maximum is taken over all D=(c_(1),c_(2),…,C_(K))∈■^(k) with 0≤c_(1)<c_(2)<…ck≤N-1.The subset A ■ ■_(N) is considered as a pseudorandom subset of degree k if P_(k)(A,N)is“small”in terms of N.We establish a link be tween the Gowers norm and our pseudorandom measure,and show that“good”pseudorandom subsets must have“small”Gowers norm.We give an example to suggest that subsets with"small" Gowers norm may have large pseudorandom measure.Finally,we prove that the pseudorandom subset of degree L(k)contains an arithmetic progression of length k,where L(k)=2·lcm(2,4,…,2|k/2|),for k≥4,and lcm(a1,a2,…,al)denotes the least common multiple of a1,a2,…,al.展开更多
基金supported in part by the National Natural Science Foundation of China(Grant No.11571277).
文摘Let A ■ ■_(N),and f_(A)(s)={1-|A/N,-|A|/N,for s ∈A,for s■ A.We define the pseudorandom measure of order k of the subset A as follows,P _(k)(A,N)=max D|∑n∈■_(N)|f_(A)(n+c_(1))f_(A)(n+c_(2))…f_(A)(n+c_(k))|where the maximum is taken over all D=(c_(1),c_(2),…,C_(K))∈■^(k) with 0≤c_(1)<c_(2)<…ck≤N-1.The subset A ■ ■_(N) is considered as a pseudorandom subset of degree k if P_(k)(A,N)is“small”in terms of N.We establish a link be tween the Gowers norm and our pseudorandom measure,and show that“good”pseudorandom subsets must have“small”Gowers norm.We give an example to suggest that subsets with"small" Gowers norm may have large pseudorandom measure.Finally,we prove that the pseudorandom subset of degree L(k)contains an arithmetic progression of length k,where L(k)=2·lcm(2,4,…,2|k/2|),for k≥4,and lcm(a1,a2,…,al)denotes the least common multiple of a1,a2,…,al.
