Approximating good simultaneous Diophantine approximations is almost NP-hard
Given a real vector α=(α1,..., α d ) and a real number ε>0 a good Diophantine approximation to α is a number Q such that ∥Qα mod ℤ∥∞ ≤ε, where ∥ · ∥∞ denotes the ℓ∞-norm ∥x∥t8 ≔ max1 ≤i≤d ¦ xi¦ for x=(x1,..., xd).
Lagarias  proved the NP-completeness of the corresponding decision problem, i.e., given a vector α ∈ ℚ d , a rational number ε>0 and a number N ∈ n+, decide whether there exists a number Q with 1 ≤Q ≤N and ¦|Qα mod ℤ∥t8≤ε.
We also investigate the nonhomogeneous variant of the good Diophantine approximation problem, i.e., given vectors α, β ∈ ℚd, a rational number ε > 0 and a number N ∈n+, decide whether there exists a number Q with 1 ≤Q ≤ N and ∥Qα-β mod ℤ∥t8, ≤ε.
This problem is particularly interesting since finding good nonhomogeneous Diophantine approximations enables us to factor integers and compute discrete logarithms (see Schnorr ).
Our results follow from recent work in the theory of probabilistically checkable proofs  and 2-prover 1-round interactive proof-systems [7, 14].
Key Wordsapproximation algorithm computational complexity NP-hard probabilistically checkable proofs Diophantine approximation 2-prover 1-round interactive proof-systems
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