General real-to-integer reductions preserving algorithmically relevant structure

Characterize when arbitrary real-valued inputs can be replaced by integer-valued inputs while preserving precisely the structural properties required by a given algorithm.

Background

The paper situates its Preprocessed 3SUM reduction within a broader effort to determine whether algorithms designed for integer inputs can be transferred to arbitrary real inputs. Integer algorithms can exploit scaling, modular hashing, and bit manipulation, whereas these techniques do not directly apply to real numbers. The paper establishes such a transfer for Preprocessed 3SUM, but does not resolve the general question of which additive or other algorithmic structures can be preserved by real-to-integer transformations.

The unresolved question concerns the precise scope of lossless or structure-preserving real-to-integer reductions: beyond the specific Preprocessed 3SUM result, it asks when integer encodings can retain exactly the relations that an algorithm needs, without imposing prohibitive preprocessing or computational overhead.