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We show here that every non-adaptive property testing algorithm making aconstant number of queries, over a fixed alphabet, can be converted to asample-based (as per [Goldreich and Ron, 2015]) testing algorithm whose averagenumber of queries is a fixed, smaller than $1$, power of $n$. Since the querydistribution of the sample-based algorithm is not dependent at all on theproperty, or the original algorithm, this has many implications in scenarioswhere there are many properties that need to be tested for concurrently, suchas testing (relatively large) unions of properties, or converting aMerlin-Arthur Proximity proof (as per [Gur and Rothblum, 2013]) to a propertesting algorithm.  The proof method involves preparing the original testing algorithm for acombinatorial analysis, which in turn involves a new result about the existenceof combinatorial structures (essentially generalized sunflowers) that allow thesample-based tester to replace the original constant query complexity tester.

Trading query complexity for sample-based testing and multi-testing scalability