State officials are looking into complaints of improper living and working conditions at public housing complexes run by the Worcester Housing Authority. The office of the Massachusetts auditor has begun a preliminary inquiry into living standards for the housing authority’s nearly 3,000 apartments. Auditor Diana DiZoglio’s office told GBH News that, depending on the findings of that review, the auditor could investigate further and look into “deficiencies, a lack of oversight or a need for greater accountability.”