Demonstrates the undecidability of extensions of Presburger arithmetic using Hardy field functions, indicating significant implications for mathematical theories.
We study the extension of Presburger arithmetic by the class of sub-polynomial Hardy field functions, and show the majority of these extensions to be undecidable. More precisely, we show that the theory Th(ℤ; < , +, ⌊f⌉), where f is a Hardy field function and ⌊⋅⌉ the nearest integer operator, is undecidable when f grows polynomially faster than x. Further, we show that when f grows sub-linearly quickly, but still as fast as some polynomial, the theory Th(ℤ; < , +, ⌊f⌉) is undecidable.
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Brown et al. (2026) studied this question.
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