In the $t$-online-erasure model in property testing, an adversary is allowed to erase $t$ values of a queried function for each query the tester makes. This model was recently formulated by Kalemaj, Raskhodnikova andVarma, who showed that the properties of linearity of functions as well as quadraticity can be tested in$O_t(1)$ many queries: $O(\log (t))$ for linearity and $2^{2^{O(t)}}$ for quadraticity. They asked whether the more general property of low-degreeness can be tested in the online erasure model, whether better testers exist for quadraticity, and if similar results hold when ``erasures'' are replaced with ``corruptions''. We show that, in the $t$-online-erasure model, for a prime power $q$, given query access to a function $f: \mathbb{F}_q^n \xrightarrow[]{} \mathbb{F}_q$, one can distinguish in $\mathrm{poly}(\log^{d+q}(t)/\delta)$ queries between the case that $f$ is degree at most $d$, and the case that $f$ is $\delta$-far from any degree $d$ function (with respect to the fractional hamming distance). This answers the aforementioned questions and brings the query complexity to nearly match the query complexity of low-degree testing in the classical property testing model. Our results are based on the observation that the property of low-degreeness admits a large and versatile family of query efficient testers. Our testers operates by querying a uniformly random, sufficiently large set of points in a large enough affine subspace, and finding a tester for low-degreeness that only utilizes queries from that set of points. We believe that this tester may find other applications to algorithms in the online-erasure model or other related models, and may be of independent interest.

翻译：在性质检验的$t$-在线擦除模型中，对手被允许在测试者每次查询时擦除被查询函数的$t$个值。该模型由Kalemaj、Raskhodnikova和Varma近期提出，他们证明了函数的线性性质以及二次性可以在$O_t(1)$次查询内完成测试：线性性质的查询复杂度为$O(\log (t))$，二次性则为$2^{2^{O(t)}}$。他们提出了以下问题：更一般的低度性质是否能在在线擦除模型中被测试？是否存在针对二次性的更优测试器？当“擦除”被替换为“损坏”时，是否仍有类似结果？我们证明，在$t$-在线擦除模型中，对于素数幂$q$，给定对函数$f: \mathbb{F}_q^n \xrightarrow[]{} \mathbb{F}_q$的查询访问，可以在$\mathrm{poly}(\log^{d+q}(t)/\delta)$次查询内区分以下两种情况：$f$的次数至多为$d$，或$f$与任意次数为$d$的函数在分数汉明距离上$\delta$-远离。这回答了上述问题，并使查询复杂度几乎匹配经典性质检验模型中低度测试的查询复杂度。我们的结果基于以下观察：低度性质允许大量且多样化的查询高效测试器。我们的测试器通过在一个足够大的仿射子空间中查询均匀随机且充分大的点集，并仅利用该点集的查询来构造低度测试器。我们相信该测试器可能为在线擦除模型或其他相关模型中的算法提供其他应用，并可能具有独立研究价值。