wp post query where authors or category with wpdb

wp post query where authors or category with wpdb

$cat_ids = "11,12,13";
$author_ids = "1,2,3";

$length = 8;
$offset = 0;

global $wpdb;

$sql = "SELECT *
        FROM $wpdb->posts
        LEFT JOIN  $wpdb->term_relationships as t
        ON ID = t.object_id
        WHERE post_type = 'post' AND post_status = 'publish'";

if ( !empty( $cat_ids ) || !empty( $author_ids ) ) :

    $sql .= " AND ( ";

    if ( !empty( $author_ids ) ) :
        $sql .= " post_author IN (".$author_ids.")";
    endif;

    if ( !empty( $cat_ids ) && !empty( $author_ids ) ) :
        $sql .= " OR "; 
    endif;

    if ( !empty( $cat_ids ) ) :
        $sql .= " term_taxonomy_id IN (".$cat_ids.")";
    endif;

    $sql .= " )";

endif;

$sql .= " ORDER BY ID DESC";

$totals = count($wpdb->get_results( $sql ));

$sql .= " LIMIT ".$length." OFFSET ".$offset."";

$results = $wpdb->get_results( $sql );

global $post;
foreach ( $results as $post ):
    setup_postdata( $post );
    ?>
    <h2><a href="<?php the_permalink(); ?>" title="<?php the_title_attribute(); ?>"><?php the_title(); ?></a></h2>            
    <?php
endforeach;

wp_reset_postdata();