Ernie Croot and Olof Sisask introduced a sampling technique for approximate translation invariance in additive combinatorics. Their Croot-Sisask almost-periodicity theorem controls how much a convolution changes under many translations.[ref]
A probabilistic technique for finding almost-periods of convolutions by Ernie Croot and Olof Sisask. Original byline and propositions 1.1 and 1.3
The useful shift is from exact algebraic symmetry to small error in a function norm. Random samples approximate averages; comparing repeated approximations exposes almost-periods. The original paper carefully specifies counting convolution and counting norms, which determine the constants and scale of the assertion.[ref]
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