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On the Behavior at Infinity of an Integrable Function

2010; Taylor & Francis; Volume: 117; Issue: 2 Linguagem: Inglês

10.4169/000298910x476095

1930-0972

Emmanuel Lesigne,

advanced mathematical theories

We prove that, in a weak sense, any integrable function on the real line tends to zero at infinity : if f is an integrable function on R, then for almost all real number x, the sequence (f(nx)) tends to zero when n goes to infinity. Using Khinchin's metric theorem on Diophantine approximation, we establish that this convergence to zero can be arbitrarily slow.