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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]

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 $L^p$ norms, which determine the constants and scale of the assertion.^[ref]

Olof Sisask co-developed the Croot-Sisask almost-periodicity theorem with Ernie Croot, introducing a probabilistic method for finding almost-periods of convolutions in additive combinatorics.^[ref]

Their proof approximates a convolution by sampling and then looks for translations preserving the approximation. Norm estimates turn that probabilistic approximation into many approximate symmetries. The method provides a way to detect additive structure without requiring exact periodicity.^[ref]