Quantitative Diophantine approximation and Fourier dimension of sets: Dirichlet non-improvable numbers versus well-approximable numbers

In this paper, we extend recent work of the third author and Ziegler on triples of integers ( a , b , c ) , with the property that each of ( a , b , c ) , ( a + 1 , b + 1 , c + 1 ) and ( a + 2 , b + 2 , c + 2 ) is multiplicatively dependent , completely classifying such triples in case a = 2 . Our techniques include a variety of elementary arguments together with more involved machinery from Diophantine approximation.

Some questions on diophantine approximation, real and p-adics

In this paper, we establish hybrid results on Diophantine approximation with primes from short intervals. In particular, we prove the following result in a slightly modified form: If [Formula: see text] is an irrational number having a continued fraction expansion with bounded terms (in particular, if [Formula: see text] is a quadratic irrational), then the number of primes p in the interval [Formula: see text] satisfying [Formula: see text] is asymptotically equal to [Formula: see text], provided that [Formula: see text], [Formula: see text] and [Formula: see text].

Abstract Theorems of Khintchine, Groshev, Jarník, and Besicovitch in Diophantine approximation are fundamental results on the metric properties of Ψ-well approximable sets. These foundational results have since been generalised to the framework of weighted Diophantine approximation for systems of real linear forms (matrices). In this article, we prove analogues of these weighted results in a range of settings including the p -adics (theorems 7 and 8), complex numbers (theorems 9 and 10), quaternions (theorems 11 and 12), and formal power series (theorems 13 and 14). The key tools in proving the main parts of these results are the weighted ubiquitous systems and weighted mass transference principle introduced recently by Kleinbock–Wang (2023 Adv. Math. ) and Wang–Wu (2021 Math. Ann ).

Counting rationals and diophantine approximation in missing-digit Cantor sets

Abstract Let and . A Diophantine tuple with property is a set of positive integers such that is a th power for all with . Such generalizations of classical Diophantine tuples have been studied extensively. In this paper, we prove several results related to robust versions of such Diophantine tuples and discuss their applications to product sets contained in a nontrivial shift of the set of all perfect powers or some of its special subsets. In particular, we substantially improve several results by Bérczes–Dujella–Hajdu–Luca, and Yip. We also prove several interesting conditional results. Our proofs are based on a novel combination of ideas from sieve methods, Diophantine approximation, and extremal graph theory.

A note on general isolation result in Diophantine approximation

Exponential shrinking problem in multiplicative Diophantine approximation

ABSTRACT It is conjectured that for any fixed relatively prime positive integers $a,b$ and c all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and z, except for specific cases. In this paper, we prove that for any fixed c there is at most one solution to the equation, except for only finitely many cases. This is regarded as a three-variable generalization of the result of Miyazaki and Pink [T. Miyazaki and I. Pink, Number of solutions to a special type of unit equations in two unknowns, III, Math. Proc. Cambridge Philos. Soc. 179 (2025), no. 3, 737–784] which asserts that for any fixed positive integer a there are only finitely many pairs of coprime positive integers b and c with $b>1$ such that the Pillai’s type equation $a^x-b^y=c$ has more than one solution in positive integers x and y. The proof of our result is based on a certain p-adic idea of Miyazaki and Pink and relies on many deep theorems on the theory of Diophantine approximation, and it also includes the complete description of solutions to some interesting system of simultaneous polynomial-exponential equations. We also discuss how effectively exceptional cases on our result for each c can be determined.

Abstract The Subspace Theorem due to Schmidt (1972) is a broad generalisation of Roth’s Theorem in Diophantine Approximation (1955) which, in the same way as the latter, suffers a notorious lack of effectivity. This problem is tackled from a probabilistic standpoint by determining the proportion of algebraic linear forms of bounded heights and degrees for which there exists a solution to the Subspace Inequality lying in a subspace of large height. The estimates are established for a class of height functions emerging from an analytic parametrisation of the projective space. They are pertinent in the regime where the heights of the algebraic quantities are larger than those of the rational solutions to the inequality under consideration, and are valid for approximation functions more general than the power functions intervening in the original Subspace Theorem. These estimates are further refined in the case of Roth’s Theorem so as to yield a Khintchin–type density version of the so–called Waldschmidt conjecture (which is known to fail pointwise). This answers a question raised by Beresnevich, Bernik and Dodson (2009).

Abstract In this survey article, we explore a central theme in Diophantine approximation inspired by a celebrated result of Besicovitch on the Hausdorff dimension of well approximable real numbers. We outline some of the key developments stemming from Besicovitch's result, with a focus on the mass transference principle, ubiquity and Diophantine approximation on manifolds and fractals. We highlight the subtle yet profound connections between number theory and fractal geometry, and discuss several open problems at their intersection.

Abstract In this paper we prove disintegration results for self-conformal measures and affinely irreducible self-similar measures. The measures appearing in the disintegration resemble self-conformal/self-similar measures for iterated function systems satisfying the strong separation condition. We use these disintegration statements to prove new results on the Diophantine properties of these measures.

Abstract In 2004, de Mathan and Teulié stated the ‐adic Littlewood conjecture (‐LC) in analogy with the classical Littlewood conjecture. Let be a finite field be an irreducible polynomial with coefficients in . This paper deals with the analogue of ‐LC over the ring of formal Laurent series over , known as the ‐adic Littlewood conjecture (‐LC). First, it is shown that any counterexample to ‐LC for the case induces a counterexample to ‐LC when is any irreducible polynomial. Since Adiceam, Nesharim and Lunnon (2021) disproved ‐LC when and when is a finite field with characteristic 3, one obtains a disproof of ‐LC over any such field in full generality (i.e., for any choice of irreducible polynomial ). The remainder of the paper is dedicated to proving two metric results on ‐LC with an additional monotonic growth function over an arbitrary finite field. The first — a Khintchine‐type theorem for ‐adic multiplicative approximation — enables one to determine the measure of the set of counterexamples to ‐LC for any choice of . The second complements this by showing that the Hausdorff dimension of the same set is maximal in the critical case where . These results are in agreement with the corresponding theory of multiplicative Diophantine approximation over the reals. Beyond the originality of the results, the main novelty of the work comes from the methodology used. Classically, Diophantine approximation employs methods from either Number Theory or Ergodic Theory. This paper provides a third option: combinatorics. Specifically, an extensive combinatorial theory is developed relating ‐LC to the properties of the so‐called number wall of a sequence. This is an infinite array containing the determinant of every finite Toeplitz matrix generated by that sequence. In full generality, the paper creates a dictionary allowing one to transfer statements in Diophantine approximation in positive characteristic to combinatorics through the concept of a number wall, and conversely.

Abstract We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to twisted diophantine approximation, and present a simple application, related to possible correlations between trace functions and dynamical sequences.

Hausdorff dimension of dynamical Diophantine approximation associated with ergodic mixing systems

Effective results in the metric theory of quantitative Diophantine approximation

Simultaneous diophantine approximation exponents of some algebraic power series in positive characteristic

We consider two questions of Ruzsa on how the minimum size of an additive basis B of a given set A depends on the domain of B. To state these questions, for an abelian group G and A⊆D⊆G we write ℓD(A):=min{|B|:B⊆D,A⊆B+B}. Ruzsa asked how much larger than ℓQ(A) can ℓZ(A) be for A⊂Z, and how much larger than ℓZ(A) can ℓN(A) be for A⊂N. For the first question we show that if ℓQ(A)=n then ℓZ(A)≤2n, and this is tight up to an additive error of at most O(n√). For the second question, we show that if ℓZ(A)=n then ℓN(A)≤O(nlogn), and this is tight up to the constant factor. We also consider these questions for higher order bases. Our proofs use some ideas that are unexpected in this context, including linear algebra and Diophantine approximation.

Uniform Diophantine approximation related to beta-transformations