# Approximating good simultaneous Diophantine approximations is almost NP-hard

## Abstract

Given a real vector *α*=(*α*_{1},..., *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} ≔ max_{1 ≤i≤d} ¦ *x*_{i}¦ for x=(*x*_{1},..., x_{d}).

Lagarias [12] 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*.

**NP⊑DTIME**(

*n*

^{poly(log n)}), there exists no polynomial-time algorithm which computes on inputs

*α*∈

*ℚ*

^{d}and

*N*∈ n

_{+}a number

*Q*

^{*}with and where γ is an arbitrary small positive constant. To put it in other words, it is almost

**NP**-hard to approximate a minimum good Diophantine approximation to

*α*in polynomial-time within a factor for an arbitrary small positive constant γ.

We also investigate the nonhomogeneous variant of the good Diophantine approximation problem, i.e., given vectors *α, β ∈* 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-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 [17]).

**NP**-complete and even approximating it in polynomial-time within a factor for an arbitrary small positive constant

*γ*is almost

**NP**-hard.

Our results follow from recent work in the theory of probabilistically checkable proofs [4] and 2-prover 1-round interactive proof-systems [7, 14].

## Key Words

approximation algorithm computational complexity**NP**-hard probabilistically checkable proofs Diophantine approximation 2-prover 1-round interactive proof-systems

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