We study the construction of the Gibbs measures for the {it focusing} mass-critical fractional nonlinear Schrodinger equation on the multi-dimensional torus. We identify the sharp mass threshold for normalizability and non-normalizability of the focusing Gibbs measures, which generalizes the influential works of Lebowitz-Rose-Speer (1988), Bourgain (1994), and Oh-Sosoe-Tolomeo (2021) on the one-dimensional nonlinear Schrodinger equations. To this purpose, we establish an almost sharp fractional Gagliardo-Nirenberg-Sobolev inequality on the torus, which is of independent interest.
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