Locality and Hilbert Curve I have a hilbert curve index based on this algorithm. I take two to four values (latitude, longitude, time in unix format and an id code) and create a 1-d hilbert curve.
I'm looking for a way to use this data to create a bounding box query (i.e. "find all ids within this rectangle).
I'm looking for a way to do so without decoding the 1d Hilbert code back into its constituent parts. 
My question is: if I created a 2d hilbert curve range (i.e. I converted the range of the box into a hilbert curve so x1y1-> hilbert value1 and x2y2-> hilbertvalue2) would all values of corresponding 2d hilbert values fall within their range?
E.g. If I converted (1,2) and (20,30) into Hilbert values and then searched for all values between hilbertvalue1 and hilbertvalue2, would all the values I get decode to fall within (1,2) and (20, 30), or would I have to perform additional transformations?
When I set all my values to 2^a* X and 2^a * y (a being a positive integer multiplier) it seems to be true. However, is there a way to use a range search on the 4d hilbert curve? I.e., if I have a Hilbert Curve made of 4 values and I have a bounding box query, can I see which hilbert values fall within that bounding box without decoding the entire Hilbert curve and checking? 
Thanks.

I just found the answer in this paper:
https://www.researchgate.net/publication/3296936_Analysis_of_the_Clustering_Properties_of_Hilbert_Space-filling_Curve
It is a dense paper, so I don't have the exact answer. But if you're motivated you can find it !
PS: you can also find the paper here
