Exponent-one logarithmic saving in the p-adic Littlewood estimate

Determine whether the exponent 1 holds in general in the logarithmic factor of the p-adic approximation bound for a real algebraic number field; specifically, establish whether, for every prime p and every basis 1, alpha_1, ..., alpha_d of a real algebraic number field K, one has liminf as n tends to infinity of n^(1/d)(log n)|n|_p max_i ||n alpha_i|| finite.

Background

Theorem 3.1 establishes a p-adic Littlewood-type estimate with logarithmic exponent 1/(r+z), where r+z is the rank of the unit group of the number field. The paper explains that this agrees with known results in degree two but is weaker than the exponent 1 suggested by earlier work and numerical evidence. The authors identify whether exponent 1 holds in general as unresolved, noting that their argument obtains the smaller exponent from the index growth of principal congruence subgroups of the unit group.

The issue is linked to finding units whose traces are divisible by large powers of p without requiring the units themselves to lie in the corresponding congruence subgroup. The authors observe that an equidistribution heuristic for traces of suitably balanced units would imply the exponent 1, and report numerical experiments consistent with that expectation.

References

We do not know whether the exponent $1$ holds in general, but numerical experiments suggest that it does; see the remark at the end of Section \ref{sec.PadicProof}.

— A geometric proof of Peck's theorem  (2609.29469 - Dhanda et al., 24 Sep 2026) in Introduction, immediately after Theorem 3.1; see also the remark at the end of Section 4 (Section \ref{sec.PadicProof})

If $n>2$, we are not able to prove that there exist integers $Q \ge 2$ such that $$ Q{1/n}\Vert Q\alp_i\Vert\ll1, \quad 1\le i \le n-1, $$ and $$ Q{1/n}\Vert Q\alp_n\Vert\log Q\ll1. $$

— Refinements of Peck's theorem on simultaneous approximation to algebraic numbers  (2609.29360 - Bugeaud et al., 24 Sep 2026) in Section 5, Additional remarks

In the case $n\ge2$, one can ask whether the results of may be refined.

— Refinements of Peck's theorem on simultaneous approximation to algebraic numbers  (2609.29360 - Bugeaud et al., 24 Sep 2026) in Section 5, Additional remarks

A number of interesting problems can be raised in connection with (3). In one direction it can be asked whether $n-1$ of the inequalities (2) can be improved with factors which are not all the same; e.g., one might conjecture that we can find infinitely many solutions of the inequalities |q_0\beta_j-q_j\beta_0|<Cq_0{-1/n}/f_j(q_0)\quad (j=1,\ldots ,n-1), |q_0\beta_n-q_n\beta_0|<Cq_0{-1/n}, with $f_1(q_0)\cdots f_{n-1}(q_0)=\log q_0$ and $f_j(q_0)\ge 1$ $(j=1,\ldots ,n-1)$.

— A geometric proof of Peck's theorem  (2609.29469 - Dhanda et al., 24 Sep 2026) in Introduction, passage quoting Peck (1961), pp. 197–198, followed by Theorems 3.2 and 3.3

Condition eqn.WeightCond is used in our proof to control the second order term in a linearization argument, and we do not know in general how to remove it.

eqn.WeightCond:

max⁡2≤i≤dfi(T) ≤ cmin⁡2≤i≤dfi(T)2,\max_{2\le i\le d}f_i(T)\ \le\ c\min_{2\le i\le d}f_i(T)^2,

— A geometric proof of Peck's theorem  (2609.29469 - Dhanda et al., 24 Sep 2026) in Introduction, immediately after Theorem 3.3 (Theorem \ref{thm.Weighted})