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Abstract

Let $ (H(n))_{n \geq 0} $ be a $2$-dimensional Halton’s sequence. Let $D_{2} ( (H(n))_{n=0}^{N-1}) $ be the $L_2$-discrepancy of $ (H_n)_{n=0}^{N-1} $. It is known that $$\limsup _{N \to \infty } (\log N)^{-1} D_{2} ( H(n) )_{n=0}^{N-1} \gt 0.$$ In this paper, we prove that $$ D_{2} (( H(n) )_{n=0}^{N-1}) =O( \log N) \quad \ \text {for} N \to \infty , $$ i.e., we find the smallest possible order of magnitude of $L_2$-discrepancy of a 2-dimensional Halton’s sequence. The main tool is theorem on linear forms in the $p$-adic logarithm.