Does Aurora DSQL support a sparse index similar to DynamoDB?

I apologize, but there seems to be a misunderstanding in your question. DSQL is not a recognized AWS service or database system that I'm aware of based on the information provided in the sources.

However, I can provide information about sparse indexes in Amazon DynamoDB, which you've correctly described. In DynamoDB, Global Secondary Indexes (GSIs) are indeed sparse by default. This means that only items in the base table that contain the specified attributes for the GSI will appear in the index.

For example, if you create a GSI with a partition key of "Status" and a sort key of "Date", only items in the base table that have both the "Status" and "Date" attributes will be included in that GSI. This feature allows for efficient querying of a subset of data without scanning the entire table.

DynamoDB's sparse indexes are particularly useful for scenarios where you need to query over a small subsection of a table, such as finding all open orders in an orders table, or identifying items that require special handling or escalation.

While I can't speak to DSQL specifically, this concept of sparse indexing is a powerful feature in DynamoDB that can help optimize query performance and reduce costs by only indexing relevant items.
Sources
Take advantage of sparse indexes - Amazon DynamoDB
Data modeling building blocks in DynamoDB - Amazon DynamoDB