Let $A=(a\_{ij})$ be an infinite doubly stochastic matrix. Does there necessarily exist a subsequence $\{n\_k\}\_{k=1}^\infty$ such that 
$$ \lim\_{k\to\infty}\frac{1}{n\_k}\sum\_{i=1}^{n\_k}\sum\_{j=1}^{n\_k}a\_{ij} >0?$$


In a previous post [A question on the partial sum of infinite doubly stochastic matrix](https://mathoverflow.net/questions/287876/a-question-on-the-partial-sum-of-infinite-doubly-stochastic-matrix), Iosif Pinelis constructed a counterexample for the whole sequence. 



On the limit of partial sum of infinite doubly stochastic matrix
No. Enumerate all positive integers which are not powers of $2$: $3=n\_1<n\_2<n\_3<\dots$ and partition positive integers into two-element sets $\{n\_k,2^{k-1}\}$. Let $a\_{i,j}=1$ if the set $\{i,j\}$ is such a two-element set, and let $a\_{i,j}=0$ otherwise. We get a symmetric bistochastic matrix, and $1$'s are only in the rows or columns indexed by a power of $2$. We have $O(\log n)$ such entries in a square $\{1,2,\dots,n\}^2$.


