For the thin obstacle problem, no homogeneous solution can have frequency in any interval (2k, 2k+1), including (2,3), in all dimensions.
Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem
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A note on the frequency gaps between integers in the thin obstacle problem
For the thin obstacle problem, no homogeneous solution can have frequency in any interval (2k, 2k+1), including (2,3), in all dimensions.