With an adaptive optics imaging survey of 148 B6-A7 stars, we have tested the hypothesis that unresolved lower-mass companions are the source of the unexpected X-ray detections of stars in this spectral type range. The sample is composed of 63 stars detected in X-rays within the ROSAT All-Sky Survey and 85 stars that form a control sample; both subsets have the same restricted distribution of spectral type, age, X-ray sensitivity and separation coverage. A total of 68 companion candidates are resolved with separations ranging from 0.3" to 26.2", with 23 new detections. The multiple star frequency of the X-ray sample based on companions resolved within the ROSAT error ellipse is found to be 43 (+6,-6)%. The corresponding control sample multiple star frequency is three times lower at 12 (+4,-3)% -- a difference of 31\pm7%. These results are presented in the first of a series of papers based on our Volume-limited A-Star (VAST) survey -- a comprehensive study of the multiplicity of A-type stars.Comment: Accepted for publication in Monthly Notices of the Royal Astronomical Societ
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