Number Theory – Diophantine Problems, Uniform Distribution and Applications

A nearly linear recurrence sequence (nlrs) is a complex sequence (a _n ) with the property that there exist complex numbers A ₀,…, A _d−1 such that the sequence \(\big(a_{n+d} + A_{d-1}a_{n+d-1} + \cdots + A_{0}a_{n}\big)_{n=0}^{\infty }\) is bounded. We give an asymptotic Binet-type formula for such sequences. We compare (a _n ) with a natural linear recurrence sequence (lrs) \((\tilde{a}_{n})\) associated with it and prove under certain assumptions that the difference sequence \((a_{n} -\tilde{ a}_{n})\) tends to infinity. We show that several finiteness results for lrs, in particular the Skolem-Mahler-Lech theorem and results on common terms of two lrs, are not valid anymore for nlrs with integer terms. Our main tool in these investigations is an observation that lrs with transcendental terms may have large fluctuations, quite different from lrs with algebraic terms. On the other hand, we show under certain hypotheses that though there may be infinitely many of them, the common terms of two nlrs are very sparse. The proof of this result combines our Binet-type formula with a Baker type estimate for logarithmic forms.

We consider a classical compound Poisson risk model with affine dividend payments. We illustrate how both by analytical and probabilistic techniques closed-form expressions for the expected discounted dividends until ruin and the Laplace transform of the time to ruin can be derived for exponentially distributed claim amounts. Moreover, numerical examples are given which compare the performance of the proposed strategy to classical barrier strategies and illustrate that such affine strategies can be a noteworthy compromise between profitability and safety in collective risk theory.

As a corollary of a general balancing result, we prove that there exists a balanced “2-coloring” g of the set of natural numbers \(\mathbb{N}\) such that simultaneously for all integers d ≥ 1, every (finite) arithmetic progression of difference d has discrepancy D _g (d) ≤ d ^8+ɛ , independently of the starting point and the length of the arithmetic progression. Formally, for every ɛ > 0 there exists a function \(g: \mathbb{N} \rightarrow \{-1,1\}\) such that for all sufficiently large d ≥ d ₀(ɛ). This reduces an old superexponential upper bound ≤ d! of Cantor, Erdős, Schreiber, and Straus to a polynomial upper bound. Note that the polynomial range is the correct range, since a well known result of Roth implies the lower bound \(D_{g}(d) \geq \sqrt{d}/20\) for every \(g: \mathbb{N} \rightarrow \{-1,1\}\).We derive this concrete number theoretic upper bound result about arithmetic progressions from a very general vector balancing result. It is about balancing infinite dimensional vectors in the maximum norm, and it is interesting in its own right (possibly, more interesting than the special case above).

We determine all integers n such that n ² has at most three base-q digits for q ∈ {2, 3, 4, 5, 8, 16}. More generally, we show that all solutions to equations of the shape where q is an odd prime, n > m > 0 and t ², | M |, N < q, either arise from “obvious” polynomial families or satisfy m ≤ 3. Our arguments rely upon Padé approximants to the binomial function, considered q-adically.

Let \(k \in \mathbb{N}\) and let f ₁, …, f _k belong to a Hardy field. We prove that under some natural conditions on the k-tuple ( f ₁, …, f _k ) the density of the set exists and equals \(\frac{1} {\zeta (k+1)}\), where ζ is the Riemann zeta function.

The theory of lacunary series starts with Weierstrass’ famous example (1872) of a continuous, nondifferentiable function and now we have a wide and nearly complete theory of lacunary subsequences of classical orthogonal systems, as well as asymptotic results for thin subsequences of general function systems. However, many applications of lacunary series in harmonic analysis, orthogonal function theory, Banach space theory, etc. require uniform limit theorems for such series, i.e., theorems holding simultaneously for a class of lacunary series, and such results are much harder to prove than dealing with individual series. The purpose of this paper is to give a survey of uniformity theory of lacunary series and discuss new results in the field. In particular, we study the permutation-invariance of lacunary series and their connection with Diophantine equations, uniform limit theorems in Banach space theory, resonance phenomena for lacunary series, lacunary sequences with random gaps, and the metric discrepancy theory of lacunary sequences.

Let X be a projective curve defined over \(\mathbb{Q}\) and \(t \in \mathbb{Q}(X)\) a non-constant rational function of degree ν ≥ 2. For every \(n \in \mathbb{Z}\) pick \(P_{n} \in X(\bar{\mathbb{Q}})\) such that t(P _n ) = n. A result of Dvornicich and Zannier implies that, for large N, among the number fields \(\mathbb{Q}(P_{1}),\ldots, \mathbb{Q}(P_{N})\) there are at least cN∕ logN distinct; here, c > 0 depends only on the degree ν and the genus g = g(X). We prove that there are at least N∕(logN)^1−η distinct fields, where η > 0 depends only on ν and g.

The distribution of differences of consecutive members of sequences of primes is investigated. A quantitative measure for oscillations among these differences is the curvature of the sequence. If the sequence is not too sparse, then sharp estimates for its curvature are provided.

Let b ≥ 2 be an integer and let s _b (n) denote the sum of the digits of the representation of an integer n in base b. For sufficiently large N, one has The proof only uses the separate (or marginal) distributions of the values of s ₂(n) and s ₃(n).

We study the discrepancy D _N of sequences \(\left (\mathbf{z}_{n}\right )_{n\geq 1} = \left (\left (\mathbf{x}_{n},y_{n}\right )\right )_{n\geq 0} \in \left [\left.0,1\right.\right )^{s+1}\) where \(\left (\mathbf{x}_{n}\right )_{n\geq 0}\) is the s-dimensional Halton sequence and \(\left (y_{n}\right )_{n\geq 1}\) is the one-dimensional Kronecker-sequence \(\left (\left \{n\alpha \right \}\right )_{n\geq 1}\). We show that for α algebraic we have \(ND_{N} = \mathcal{O}\left (N^{\varepsilon }\right )\) for all ɛ > 0. On the other hand, we show that for α with bounded continued fraction coefficients we have \(ND_{N} = \mathcal{O}\left (N^{\frac{1} {2} }(\log N)^{s}\right )\) which is (almost) optimal since there exist α with bounded continued fraction coefficients such that \(ND_{N} = \Omega \left (N^{\frac{1} {2} }\right )\).

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple, and Dujella, Kazalicki, Mikić and Szikszai recently proved that there exist infinitely many rational Diophantine sextuples.

In this paper, generalizing the work of Piezas, we describe a method for generating new parametric formulas for rational Diophantine sextuples.

Let A be an integral domain with quotient field K of characteristic 0 that is finitely generated as a \(\mathbb{Z}\)-algebra. Denote by D(F) the discriminant of a polynomial F ∈ A[X]. Further, given a finite étale K-algebra \(\Omega\), denote by \(D_{\Omega /K}(\alpha )\) the discriminant of α over K. For non-zero δ ∈ A, we consider equations to be solved in monic polynomials F ∈ A[X] of given degree n ≥ 2 having their zeros in a given finite extension field G of K, and where O is an A-order of \(\Omega\), i.e., a subring of the integral closure of A in \(\Omega\) that contains A as well as a K-basis of \(\Omega\).

In the series of papers (Győry, Acta Arith 23:419–426, 1973; Győry, Publ Math Debrecen 21:125–144, 1974; Győry, Publ Math Debrecen 23:141–165, 1976; Győry, Publ Math Debrecen 25:155–167, 1978; Győry, Acta Math Acad Sci Hung 32:175–190, 1978; Győry, J Reine Angew Math 324:114–126, 1981), Győry proved that when K is a number field, A the ring of integers or S-integers of K, and \(\Omega\) a finite field extension of K, then up to natural notions of equivalence the above equations have, without fixing G, finitely many solutions, and that moreover, if K, S, \(\Omega\), O, and δ are effectively given, a full system of representatives for the equivalence classes can be effectively determined. Later, Győry (Publ Math Debrecen 29:79–94, 1982) generalized in an ineffective way the above-mentioned finiteness results to the case when A is an integrally closed integral domain with quotient field K of characteristic 0 which is finitely generated as a \(\mathbb{Z}\)-algebra and G is a finite extension of K. Further, in Győry (J Reine Angew Math 346:54–100, 1984) he made these results effective for a special class of integral domains A containing transcendental elements. In Evertse and Győry (Discriminant equations in diophantine number theory, Chap. 10 Cambridge University Press, 2016) we generalized in an effective form the results of Győry (Publ Math Debrecen 29:79–94, 1982) mentioned above to the case where A is an arbitrary integrally closed domain of characteristic 0 which is finitely generated as a \(\mathbb{Z}\)-algebra, where \(\Omega\) is a finite étale K-algebra, and where A, δ, and G, respectively \(\Omega,O\) are effectively given (in a well-defined sense described below).

Let P be a polynomial that depends on two variables X and Y and has algebraic coefficients. If x and y are algebraic numbers with P(x, y) = 0, then by work of Néron h(x)∕q is asymptotically equal to h(y)∕p where p and q are the partial degrees of P in X and Y, respectively. In this paper we compute a completely explicit bound for | h(x)∕q − h(y)∕p | in terms of P which grows asymptotically as max{h(x), h(y)}^1∕2. We apply this bound to obtain a simple version of Runge’s Theorem on the integral solutions of certain polynomial equations.

A Lucas sequence is a sequence of the general form \(v_{n} = (\phi ^{n} -\overline{\phi }^{n})/(\phi -\overline{\phi })\), where ϕ and \(\overline{\phi }\) are real algebraic integers such that \(\phi +\overline{\phi }\) and \(\phi \overline{\phi }\) are both rational. Famous examples include the Fibonacci numbers, the Pell numbers, and the Mersenne numbers. We study the monoid that is generated by such a sequence; as it turns out, it is almost freely generated. We provide an asymptotic formula for the number of positive integers ≤ x in this monoid, and also prove Erdős–Kac type theorems for the distribution of the number of factors, with and without multiplicity. While the limiting distribution is Gaussian if only distinct factors are counted, this is no longer the case when multiplicities are taken into account.

We prove a relation between two measures of pseudorandomness, the arithmetic autocorrelation, and the correlation measure of order k. Roughly speaking, we show that any binary sequence with small correlation measure of order k up to a sufficiently large k cannot have a large arithmetic correlation. We apply our result to several classes of sequences including Legendre sequences defined with polynomials.

In the present paper we are interested in number systems in the ring of integers of cyclotomic number fields in order to obtain a result equivalent to Cobham’s theorem. For this reason we first search for potential bases. This is done in a very general way in terms of canonical number systems. In a second step we analyse pairs of bases in view of their multiplicative independence. In the last part we state an appropriate variant of Cobham’s theorem.

This article deals with estimates for single exponential sums, combining tools from the classic Van der Corput’s theory with an ingredient from M. Huxley’s work. Further, a very precise way of balancing terms is applied with gain. The result obtained is used to derive asymptotic estimates for the coefficients of products of three Dirichlet L-series, as was initiated by Friedlander and Iwaniec (Can. J. Math. 57(3):494–505, 2005).

We study intersections of orbits in polynomial dynamics with multiplicative subgroups and subfields of arbitrary fields of characteristic zero, as well as with sets of points that are close with respect to the Weil height to division groups of finitely generated groups of \(\overline{\mathbb{Q}}^{{\ast}}\).

After the proof of Zhang about the existence of infinitely many bounded gaps between consecutive primes the author showed the existence of a bounded d such that there are arbitrarily long arithmetic progressions of primes with the property that p ^′ = p + d is the prime following p for each element of the progression. This was a common generalization of the results of Zhang and Green-Tao. In the present work it is shown that for every m we have a bounded m-tuple of primes such that this configuration (i.e. the integer translates of this m-tuple) appear as arbitrarily long arithmetic progressions in the sequence of all primes. In fact we show that this is true for a positive proportion of all m-tuples. This is a common generalization of the celebrated works of Green-Tao and Maynard/Tao.

It is proved that every simple linear recurrence defined over a number field K, that has zeros modulo almost all prime ideals of K, takes the value 0 for a certain integer index. A similar theorem does not hold, in general, for simple linear recurrences of order n > 3. The case n = 3 is studied, but not decided.

Let λ denote the Liouville function. The Chowla conjecture asserts that for any fixed natural numbers \(a_{1},a_{2},\ldots,a_{k}\) and non-negative integer \(b_{1},b_{2},\ldots,b_{k}\) with \(a_{i}b_{j} - a_{j}b_{i}\neq 0\) for all \(1\leqslant i <j\leqslant k\), and any \(X\geqslant 1\). This conjecture is open for \(k\geqslant 2\). As is well known, this conjecture implies the conjecture of Sarnak that whenever \(f: \mathbb{N} \rightarrow \mathbb{C}\) is a fixed deterministic sequence and \(X\geqslant 1\). In this paper, we consider the weaker logarithmically averaged versions of these conjectures, namely that and under the same hypotheses on \(a_{1},\ldots,a_{k},b_{1},\ldots,b_{k}\) and f, and for any \(2\leqslant \omega \leqslant X\). Our main result is that these latter two conjectures are logically equivalent to each other, as well as to the “local Gowers uniformity” of the Liouville function. The main tools used here are the entropy decrement argument of the author used recently to establish the k = 2 case of the logarithmically averaged Chowla conjecture, as well as the inverse conjecture for the Gowers norms, obtained by Green, Ziegler, and the author.

Van der Corput and Halton sequences are well-known low-discrepancy sequences. Almost 20 years ago Ninomiya defined analogues of van der Corput sequences for β-numeration and proved that they also form low-discrepancy sequences if β is a Pisot number. Only very recently Robert Tichy and his co-authors succeeded in proving that \(\boldsymbol{\beta }\)-adic Halton sequences are equidistributed for certain parameters \(\boldsymbol{\beta }= (\beta _{1},\ldots,\beta _{s})\) using methods from ergodic theory. In the present paper we continue this research and give discrepancy estimates for \(\boldsymbol{\beta }\)-adic Halton sequences for which the components β _i are m-bonacci numbers. Our methods are quite different and use dynamical and geometric properties of Rauzy fractals that allow to relate \(\boldsymbol{\beta }\)-adic Halton sequences to rotations on high dimensional tori. The discrepancies of these rotations can then be estimated by classical methods relying on W.M. Schmidt’s Subspace Theorem.