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If $g\in L^p(\mathbb{R}^n)$ and $1\leq p<\infty$ then show $$\lim_{|t|\to \infty}\lVert g_{(t)}+g\rVert_p=2^{1/p}\lVert g\rVert_p,$$

where $g_{(t)}(x):=g(t+x)$.

Any hints? Try to give me only hints/outlines not complete solutions

Not sure where to go from there?

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    You can try to prove it for characteristic function of hypercube , and then for characteristic function of any set, with using iterated limits, starting from t. An example of using iterated limits is here: http://math.stackexchange.com/questions/677108/proof-of-riemann-lebesgue-lemma/711617#711617 ,you can also consider inequality described here: http://math.stackexchange.com/questions/458230/continuity-of-l1-functions-with-respect-to-translation/458235#458235, or some tricks from this document: http://fractal.math.unr.edu/~ejolson/761/notes/761sep12.pdf2014-04-23

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