I use REST web service for the purpose.
The official guide is too obscure. Below you can find samples that cover simple and advanced queries. They will make things clear:
full-text and property "title"
_https://MYTENANT.sharepoint.com/_api/search/query?QueryText='part title:document'&selectproperties='Title,Author,Size,Path,Description,Write,LastModifiedTime,HitHighlightedSummary,ServerRedirectedPreviewURL,FileExtension,UniqueId,DisplayAuthor,UrlZone,Culture'&rowlimit=20&rowsperpage=20
full-text and property "author" and "title"
_https://MYTENANT.sharepoint.com/_api/search/query?QueryText='part title:document author:kapil'&selectproperties='Title,Author,Size,Path,Description,Write,LastModifiedTime,HitHighlightedSummary,ServerRedirectedPreviewURL,FileExtension,UniqueId,DisplayAuthor,UrlZone,Culture'&rowlimit=20&rowsperpage=20
full-text and size limitation
_https://MYTENANT.sharepoint.com/_api/search/query?QueryText='part title:document size>60000'&selectproperties='Title,Author,Size,Path,Description,Write,LastModifiedTime,HitHighlightedSummary,ServerRedirectedPreviewURL,FileExtension,UniqueId,DisplayAuthor,UrlZone,Culture'&rowlimit=20&rowsperpage=20
full-text and last modified time
_https://MYTENANT.sharepoint.com/_api/search/query?QueryText='part title:document LastModifiedTime>2014-08-04T04:09:00.0000000Z'&selectproperties='Title,Author,Size,Path,Description,Write,LastModifiedTime,HitHighlightedSummary,ServerRedirectedPreviewURL,FileExtension,UniqueId,DisplayAuthor,UrlZone,Culture'&rowlimit=20&rowsperpage=20
full-text and last modified date
_https://MYTENANT.sharepoint.com/_api/search/query?QueryText='part title:document LastModifiedTime>2014-08-04'&selectproperties='Title,Author,Size,Path,Description,Write,LastModifiedTime,HitHighlightedSummary,ServerRedirectedPreviewURL,FileExtension,UniqueId,DisplayAuthor,UrlZone,Culture'&rowlimit=20&rowsperpage=20