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Abstract

We extend the results on uniform distribution modulo 1 given in [3] to sequences of the form $(t(h_nF(n{\mit \Theta })+\varepsilon _nh'_n))_n$, where $(h_n)_n$, $(h'_n)_n$ and $(h_n/h'_n)_n$ are polynomially increasing sequences, $(\varepsilon _n)_n$ a bounded sequence, $F$~: ${\mathbb R}\rightarrow {\mathbb R}$ essentially a $1$-periodic $C^3$ function, ${\mit \Theta }$ and $t$ real numbers (the case $F$~: ${\mathbb R}^d\rightarrow {\mathbb R}$ and ${\mit \Theta }\in {\mathbb R}^d$ for $d>1$ will be treated in a separate article). We remove the diophantine hypothesis on ${\mit \Theta }$ needed in [3], and add a technical hypothesis on $h_n$.