  With a combination of adaptive optics imaging and a multi-epoch common proper
motion search, we have conducted a large volume-limited (D $\le$ 75 pc)
multiplicity survey of A-type stars, sensitive to companions beyond 30 au. The
sample for the Volume-limited A-STar (VAST) survey consists of 435 A-type
stars: 363 stars were observed with adaptive optics, 228 stars were searched
for wide common proper motion companions and 156 stars were measured with both
techniques. The projected separation coverage of the VAST survey extends from
30 to 45,000 au. A total of 137 stellar companions were resolved, including 64
new detections from the VAST survey, and the companion star fraction, projected
separation distribution and mass ratio distribution were measured. The
separation distribution forms a log-normal distribution similar to the
solar-type binary distribution, but with a peak shifted to a significantly
wider value of 387 (+132,-98) au. Integrating the fit to the distribution over
the 30 to 10,000 au observed range, the companion star fraction for A-type
stars is estimated as 33.8%+-2.6%. The mass ratio distribution of closer (<125
au) binaries is distinct from that of wider systems, with a flat distribution
for close systems and a distribution that tends towards smaller mass ratios for
wider binaries. Combining this result with previous spectroscopic surveys of
A-type stars gives an estimate of the total companion star fraction of
68.9%+-7.0%. The most complete assessment of higher order multiples was
estimated from the 156-star subset of the VAST sample with both adaptive optics
and common proper motion measurements, combined with a literature search for
companions, yielding a lower limit on the frequency of single, binary, triple,
quadruple and quintuple A-type star systems of 56.4 (-4.0,+3.8), 32.1
(-3.5,+3.9), 9.0 (-1.8,+2.8), 1.9 (-0.6,+1.8) and 0.6 (-0.2,+1.4) per cent,
respectively.
