We consider the problem of covering $\mathbb{Z}^2$ with a finite number of sublattices of finite index; this may be viewed as a projective, or homogeneous, version of the well-known problem of covering systems of congruences. We give a construction of minimal coverings which produces many, but not all, minimal coverings, prove an inequality relating the size of a covering to the least common multiple of the lattice indices, and determine all minimal coverings with at most $8$ sublattices.
We consider a system of homogeneous quadratic forms with congruence conditions in $n\geq 3$ variables and prove the existence of two linearly independent integral solutions of bounded height. We also show the existence of small height integral zeros of this system avoiding a given set of hyperplanes.
Sieving by large integers and covering systems of congruences
一个老问题的鄂尔多斯问，如果存在，对每个数N，一个有限集S的整数大于N和剩余类r（n）mod n n n n在S的工会是所有的整数。证明了如果$\sum_{n\in S} 1/n$对这样的整数覆盖是有界的，则S的最小成员也是有界的，从而证实了Erdos和Selfridge的一个猜想.我们还证明了Erdos和Graham的一个猜想，即对每个固定数K>1，剩余类r（n）mod n的任何并集的整数中的补数，对于（N，KN]中的不同n，当N足够大时，密度至少为d_K。这里d_K是一个仅取决于K的正数。这两个新的结果都暗示了Erdos和Graham的另一个猜想，即如果S是一个模大于大于N的有限集，且对覆盖整数的S中的n有剩余类r（n）mod n的选择，则S的最大成员不可能是O（N）。我们进一步得到这些结果的更强的形式，并建立其他信息，包括Haight的相关定理的改进。
Small solutions to inhomogeneous and homogeneous quadratic congruences modulo prime powers
Ramakrishna Mission Vivekananda Educational and Research Institute（拉马克里希纳传教士维韦卡南达教育与研究所）；Department of Mathematics, A. P. C. Roy Government College（A.P.C.罗伊政府学院数学系）
