Around Furstenberg's times $p$, times $q$ conjecture: times $p$-invariant measures with some large Fourier coefficients

For each integer $n\ge 1$, denote by $T_n$ the map $x\mapsto nx\mathrm{m}\mathrm{o}\mathrm{d}1$ from the circle group $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$ into itself. Let $p,q\ge 2$ be two multiplicatively independent integers. Using Baire Category arguments, we show that generically a $T_p$-invariant probability measure $\mu$ on $\mathbb{T}$ with no atom has some large Fourier coefficients along the sequence $(q^n{)}_{n\ge 0}$. In particular, $(T_{q^n}\mu {)}_{n\ge 0}$ does not converges weak-star to the normalised Lebesgue measure on $\mathbb{T}$. This disproves a conjecture of Furstenberg and complements previous results of Johnson and Rudolph. In the spirit of previous work by Meiri and Lindenstrauss-Meiri-Peres, we study generalisations of our main result to certain classes of sequences $(c_n{)}_{n\ge 0}$ other than the sequences $(q^n{)}_{n\ge 0}$, and also investigate the multidimensional setting.