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We show that if $\\mu$ is a probability measure on $[0,1)$ such that $|\\widehat{\\mu}(t)|\\leq c|t|^{-\\eta}$, then for $\\mu$-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies \\begin{equation*} \\frac{1}{4} \\leq \\limsup_{N\\to\\infty}\\frac{N D_N(n_kx)}{\\sqrt{N\\log\\log N}} \\leq C \\end{equation*} for some constant $C>0$, proving a conjecture of Haynes, Jensen and Kristensen. 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