A well-known result of Verstraëte [23] shows that for each integer $k\ge 2$ every graph G with average degree at least 8k contains cycles of k consecutive even lengths, the shortest of which is of length at most twice the radius of G. We establish two extensions of Verstraëte's result for linear cycles in linear r-uniform hypergraphs.

We show that for any fixed integers $r\ge 3$ and $k\ge 2$, there exist constants $c_1=c_1(r)$ and $c_2=c_2(r)$, such that every n-vertex linear r-uniform hypergraph G with average degree $d(G)\ge c_{1}k$ contains linear cycles of k consecutive even lengths, the shortest of which is of length at most $2\lceil \frac{\log n}{\log (d(G)/k)-c_2}\rceil$. In particular, as an immediate corollary, we retrieve the current best known upper bound on the linear Turán number of $C_{2k}^r$ with improved coefficients.

Furthermore, we show that for any fixed integers $r\ge 3$ and $k\ge 2$, there exist constants $c_3=c_3(r)$ and $c_4=c_4(r)$ such that every n-vertex linear r-uniform hypergraph with average degree $d(G)\ge c_{3}k$, contains linear cycles of k consecutive lengths, the shortest of which has length at most $6\lceil \frac{\log n}{\log (d(G)/k)-c_4}\rceil +6$. In both cases for given average degree d, the length of the shortest cycles cannot be improved up to the constant factors $c_2,c_4$.

Verstraëte [23]的一个众所周知的结果表明，对于每个整数$k\ge 2$每个平均度数至少为 8 k的图G都包含k 个连续偶数长度的循环，其中最短的循环的长度最多为G半径的两倍。我们建立了 Verstraëte 在线性r均匀超图中的线性循环结果的两个扩展。

我们证明对于任何固定整数$r\ge 3$和$k\ge 2$, 存在常数$C_1=C_1（r）$和$C_2=C_2（r）$，使得每个n顶点线性r均匀超图G具有平均度$d（G）\ge C_{1}k$包含k 个连续偶数长度的线性循环，其中最短的长度最多为$2\lceil \frac{\mathrm{日志}n}{\mathrm{日志}（d（G）/k）-C_2}\rceil$。特别是，作为直接推论，我们检索了当前最著名的线性图兰数上限$C_{2k}^r$具有改进的系数。

此外，我们证明对于任何固定整数$r\ge 3$和$k\ge 2$, 存在常数$C_3=C_3（r）$和$C_4=C_4（r）$使得每个n顶点线性r均匀超图具有平均度$d（G）\ge C_{3}k$，包含k 个连续长度的线性循环，其中最短的长度最多$6\lceil \frac{\mathrm{日志}n}{\mathrm{日志}（d（G）/k）-C_4}\rceil +6$。在这两种情况下，对于给定的平均度d，最短周期的长度不能提高到常数因子$C_2,C_4$。