Context :

Aims : We search for low-mass companions of M dwarfs and characterize their multiplicity fraction with the purpose of helping in the selection of the most appropriate targets for the CARMENES exoplanet survey .

Methods : We obtained high-resolution images in the I band with the lucky imaging instrument FastCam at the 1.5 m Telescopio Carlos Sánchez for 490 mid- to late-M dwarfs .
For all the detected binaries , we measured angular separations , position angles , and magnitude differences in the I band .
We also calculated the masses of each individual component and estimated orbital periods , using the available magnitude and colour relations for M dwarfs and our own M _ { J } -spectral type and mass- M _ { I } relations .
To avoid biases in our sample selection , we built a volume-limited sample of M0.0-M5.0 dwarfs that is complete up to 86 % within 14 pc .

Results : From the 490 observed stars , we detected 80 companions in 76 systems , of which 30 are new discoveries .
Another six companion candidates require additional astrometry to confirm physical binding .
The multiplicity fraction in our observed sample is 16.7 \pm 2.0 % .
The bias-corrected multiplicity fraction in our volume-limited sample is 19.5 \pm 2.3 % for angular separations of 0.2 to 5.0 arcsec ( 1.4–65.6 au ) , with a peak in the distribution of the projected physical separations at 2.5–7.5 au .
For M0.0-M3.5 V primaries , our search is sensitive to mass ratios higher than 0.3 and there is a higher density of pairs with mass ratios over 0.8 compared to those at lower mass ratios .
Binaries with projected physical separations shorter than 50 au also tend to be of equal mass .
For 26 of our systems , we estimated orbital periods shorter than 50 a , 10 of which are presented here for the first time .
We measured variations in angular separation and position angle that are due to orbital motions in 17 of these systems .
The contribution of binaries and multiples with angular separations shorter than 0.2 arcsec , longer than 5.0 arcsec , and of spectroscopic binaries identified from previous searches , although not complete , may increase the multiplicity fraction of M dwarfs in our volume-limited sample to at least 36 % .

Conclusions :