We examine the spectra of 23 fast radio bursts detected in a flys-eye survey with the Australian SKA Pathfinder, including those of three bursts not previously reported. The mean spectral index of $alpha = -1.6_{-0.2}^{+0.3}$ ($F_ u propto u^alpha$) is close to that of the Galactic pulsar population. The sample is dominated by bursts exhibiting a large degree of spectral modulation: 17 exhibit fine-scale spectral modulation with an rms exceeding 50% of the mean, with decorrelation bandwidths (half-maximum) ranging from $approx$ to 49 MHz. Most decorrelation bandwidths are an order of magnitude lower than the $gtrsim 30,$MHz expected of Galactic interstellar scintillation at the Galactic latitude of the survey, $|b| = 50 pm 5 deg$. A test of the amplitude distribution of the spectral fluctuations reveals only 12 bursts consistent at better than a 5% confidence level with the prediction of 100%-modulated diffractive scintillation. Moreover, five of six FRBs with a signal-to-noise ratio exceeding 18 are consistent with this prediction at less than 1% confidence. Nonetheless, there is weak evidence (88-95% confidence) that the amplitude of the fine-scale spectral modulation is anti-correlated with dispersion measure (DM) that would suggest it originates as a propagation effect. This effect appears to be corroborated by the smoothness of the higher-DM Parkes FRBs, and could arise due to quenching of diffractive scintillation (e.g. in the interstellar medium of the host galaxy) by angular broadening in the intergalactic medium.
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