Conjecture 2.5 — Lang-Waldschmidt

Definitions and mathematical statement

Fix a real $\varepsilon>0$. For any integer $m\ge1$, choose nonzero integers $a_1,\ldots,a_m$ and nonzero integer exponents $b_1,\ldots,b_m$. Negative bases and negative exponents are allowed; integer powers therefore define an unambiguous nonzero rational number \[ A=\prod_{i=1}^m a_i^{b_i}. \] Write $B=\max_i|b_i|$, and suppose $A\ne1$.

The Lang--Waldschmidt conjecture in this multiplicative formulation asks whether there is a constant $C(\varepsilon)>0$, depending only on $\varepsilon$, such that every such choice satisfies \[ |A-1|\ge \frac{C(\varepsilon)^m B} {\left(\prod_{i=1}^m|b_i|\prod_{i=1}^m|a_i|\right)^{1+\varepsilon}}. \] The constant must be chosen before $m$, the bases, and the exponents. Its allowed dependence on the number of factors is precisely through the displayed power $C(\varepsilon)^m$.

No coprimality or multiplicative-independence assumption is imposed on the bases. The sole exclusion $A=1$ is necessary, since otherwise the left side would vanish while the denominator and numerator remain positive. Exponents are required nonzero to keep the formula defined; factors with zero exponent can simply be omitted before applying the statement.

The problem is a uniform quantitative lower bound for a nontrivial rational multiplicative expression near one. Establishing a positive bound that depends arbitrarily on the individual bases or exponents would not give the asserted dependence.

Short English statement

Is every nontrivial product of integer powers bounded away from one by the stated uniform Lang–Waldschmidt estimate?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 8700013 (AMR-086-0013), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Waldschmidt - Open Diophantine Problems (2004), Conjecture 2.5. Original source indexed by AMR-086-0013. https://arxiv.org/abs/math/0312440.