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Let K be an algebraic number field. We investigate the K-rational distance problem and prove that there are infinitely many nonisomorphic cubic number fields and a number field of degree n for every
$n\geq 2$
in which there is a point in the plane of a unit square at K-rational distances from the four vertices of the square.
Aigner showed in 1934 that nontrivial quadratic solutions to
$x^4 + y^4 = 1$
exist only in
$\mathbb Q(\sqrt {-7})$
. Following a method of Mordell, we show that nontrivial quadratic solutions to
$x^4 + 2^ny^4 = 1$
arise from integer solutions to the equations
$X^4 \pm 2^nY^4 = Z^2$
investigated in 1853 by V. A. Lebesgue.
Jeśmanowicz conjectured that
$(x,y,z)=(2,2,2)$
is the only positive integer solution of the equation
$(*)\; ((\kern1.5pt f^2-g^2)n)^x+(2fgn)^y=((\kern1.5pt f^2+g^2)n)^x$
, where n is a positive integer and f, g are positive integers such that
$f>g$
,
$\gcd (\kern1.5pt f,g)=1$
and
$f \not \equiv g\pmod 2$
. Using Baker’s method, we prove that: (i) if
$n>1$
,
$f \ge 98$
and
$g=1$
, then
$(*)$
has no positive integer solutions
$(x,y,z)$
with
$x>z>y$
; and (ii) if
$n>1$
,
$f=2^rs^2$
and
$g=1$
, where r, s are positive integers satisfying
$(**)\; 2 \nmid s$
and
$s<2^{r/2}$
, then
$(*)$
has no positive integer solutions
$(x,y,z)$
with
$y>z>x$
. Thus, Jeśmanowicz’ conjecture is true if
$f=2^rs^2$
and
$g=1$
, where r, s are positive integers satisfying
$(**)$
.
We show that the set of natural numbers has an exponential diophantine definition in the rationals. It follows that the corresponding decision problem is undecidable.
We sharpen earlier work of Dabrowski on near-perfect power values of the quartic form $x^{4}-y^{4}$, through appeal to Frey curves of various signatures and related techniques.
Erdös and Zaremba showed that
$ \limsup_{n\to \infty} \frac{\Phi(n)}{(\log\log n)^2}=e^\gamma$
, γ being Euler’s constant, where
$\Phi(n)=\sum_{d|n} \frac{\log d}{d}$
.
We extend this result to the function
$\Psi(n)= \sum_{d|n} \frac{(\log d )(\log\log d)}{d}$
and some other functions. We show that
$ \limsup_{n\to \infty}\, \frac{\Psi(n)}{(\log\log n)^2(\log\log\log n)}\,=\, e^\gamma$
. The proof requires a new approach. As an application, we prove that for any
$\eta>1$
, any finite sequence of reals
$\{c_k, k\in K\}$
,
$\sum_{k,\ell\in K} c_kc_\ell \, \frac{\gcd(k,\ell)^{2}}{k\ell} \le C(\eta) \sum_{\nu\in K} c_\nu^2(\log\log\log \nu)^\eta \Psi(\nu)$
, where C(η) depends on η only. This improves a recent result obtained by the author.
We show that any smooth projective cubic hypersurface of dimension at least 29 over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley.
We include an appendix in which we highlight some slight modifications to a recent result of Papanikolopoulos and Siksek. It follows that the set of rational points on smooth projective cubic hypersurfaces of dimension at least 29 is generated via secant and tangent constructions from just a single point.
We show, under some natural restrictions, that some semigroup orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime
$p$
, extending previous work of Shparlinski [‘Multiplicative orders in orbits of polynomials over finite fields’, Glasg. Math. J.60(2) (2018), 487–493].
Let
$\mathbb{Z}$
and
$\mathbb{Z}^{+}$
be the set of integers and the set of positive integers, respectively. For
$a,b,c,d,n\in \mathbb{Z}^{+}$
, let
$t(a,b,c,d;n)$
be the number of representations of
$n$
by
$\frac{1}{2}ax(x+1)+\frac{1}{2}by(y+1)+\frac{1}{2}cz(z+1)+\frac{1}{2}dw(w+1)$
with
$x,y,z,w\in \mathbb{Z}$
. Using theta function identities we prove 13 transformation formulas for
$t(a,b,c,d;n)$
and evaluate
$t(2,3,3,8;n)$
,
$t(1,1,6,24;n)$
and
$t(1,1,6,8;n)$
.
We study the generalized Fermat equation
$x^{2}+y^{3}=z^{p}$
, to be solved in coprime integers, where
$p\geqslant 7$
is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve
$X(p)$
. We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic
$p$
-torsion modules. Using these criteria we produce the minimal list of twists of
$X(p)$
that have to be considered, based on local information at 2 and 3; this list depends on
$p\hspace{0.2em}{\rm mod}\hspace{0.2em}24$
. We solve the equation completely when
$p=11$
, which previously was the smallest unresolved
$p$
. One new ingredient is the use of the ‘Selmer group Chabauty’ method introduced by the third author, applied in an elliptic curve Chabauty context, to determine relevant points on
$X_{0}(11)$
defined over certain number fields of degree 12. This result is conditional on the generalized Riemann hypothesis, which is needed to show correctness of the computation of the class groups of five specific number fields of degree 36. We also give some partial results for the case
$p=13$
. The source code for the various computations is supplied as supplementary material with the online version of this article.
We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$, where $\unicode[STIX]{x1D6E5}$ is a finite-index subgroup of $\text{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$, where for all $j$, $\mathbf{a}_{j}$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\text{SL}(n+1,\mathbb{R})$ developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form $\mathbf{A}$.
Let $X$ be a finite-dimensional connected compact abelian group equipped with the normalized Haar measure $\unicode[STIX]{x1D707}$. We obtain the following mean ergodic theorem over ‘thin’ phase sets. Fix $k\geq 1$ and, for every $n\geq 1$, let $A_{n}$ be a subset of $\mathbb{Z}^{k}\cap [-n,n]^{k}$. Assume that $(A_{n})_{n\geq 1}$ has $\unicode[STIX]{x1D714}(1/n)$ density in the sense that $\lim _{n\rightarrow \infty }(|A_{n}|/n^{k-1})=\infty$. Let $T_{1},\ldots ,T_{k}$ be ergodic automorphisms of $X$. We have
for any $f_{1},\ldots ,f_{k}\in L_{\unicode[STIX]{x1D707}}^{\infty }$. When the $T_{i}$ are ergodic epimorphisms, the same conclusion holds under the further assumption that $A_{n}$ is a subset of $[0,n]^{k}$ for every $n$. The density assumption on the $A_{i}$ is necessary. Immediate applications include certain Poincaré style recurrence results.
Such a sequence is eventually periodic and we denote by $P(n)$ the maximal period of such sequences for given $n$. We prove a new upper bound in the case where $n$ is a power of a prime $p\equiv 5\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ for which $2$ is a primitive root and the Pellian equation $x^{2}-py^{2}=-4$ has no solutions in odd integers $x$ and $y$.
Let $D$ be a positive nonsquare integer, $p$ a prime number with $p\nmid D$ and $0<\unicode[STIX]{x1D70E}<0.847$. We show that there exist effectively computable constants $C_{1}$ and $C_{2}$ such that if there is a solution to $x^{2}+D=p^{n}$ with $p^{n}>C_{1}$, then for every $x>C_{2}$ with $x^{2}+D=p^{n}m$ we have $m>x^{\unicode[STIX]{x1D70E}}$. As an application, we show that for $x\neq \{5,1015\}$, if the equation $x^{2}+76=101^{n}m$ holds, then $m>x^{0.14}$.
We generalize work by Bourgain and Kontorovich [On the local-global conjecture for integral Apollonian gaskets, Invent. Math. 196 (2014), 589–650] and Zhang [On the local-global principle for integral Apollonian 3-circle packings, J. Reine Angew. Math. 737, (2018), 71–110], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group
${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$
satisfying certain conditions, where
$K$
is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof is that
${\mathcal{A}}$
possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in
$\operatorname{PSL}_{2}({\mathcal{O}}_{K})$
containing a Zariski dense subgroup of
$\operatorname{PSL}_{2}(\mathbb{Z})$
.
The Chabauty–Kim method allows one to find rational points on curves under certain technical conditions, generalising Chabauty’s proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus. We show how the Chabauty–Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a non-abelian generalisation of Coleman’s effective Chabauty theorem.
In this paper we show that a polynomial equation admits infinitely many prime-tuple solutions, assuming only that the equation satisfies suitable local conditions and the polynomial is sufficiently non-degenerate algebraically. Our notion of algebraic non-degeneracy is related to the
$h$
-invariant introduced by W. M. Schmidt. Our results prove a conjecture by B. Cook and Á. Magyar for hypersurfaces of degree 3.
We prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed m there are at most
${\cal O}_{\epsilon }(n^{{3}/{5}+\epsilon })$
solutions of
${m}/{n} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$
. This improves upon a result of Browning and Elsholtz (2011) and extends a result of Elsholtz and Tao (2013) who proved this when m=4 and n is a prime. Moreover, there exists an algorithm finding all solutions in expected running time
${\cal O}_{\epsilon }(n^{\epsilon }({n^3}/{m^2})^{{1}/{5}})$
, for any
$\epsilon \gt 0$
. We also improve a bound on the maximum number of representations of a rational number as a sum of k unit fractions. Furthermore, we also improve lower bounds. In particular, we prove that for given
$m \in {\open N}$
in every reduced residue class e mod f there exist infinitely many primes p such that the number of solutions of the equation
${m}/{p} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$
is
$\gg _{f,m} \exp (({5\log 2}/({12\,{\rm lcm} (m,f)}) + o_{f,m}(1)) {\log p}/{\log \log p})$
. Previously, the best known lower bound of this type was of order
$(\log p)^{0.549}$
.
We show that Hermite’s theorem fails for every integer $n$ of the form $3^{k_{1}}+3^{k_{2}}+3^{k_{3}}$ with integers $k_{1}>k_{2}>k_{3}\geqslant 0$. This confirms a conjecture of Brassil and Reichstein. We also obtain new results for the relative Hermite–Joubert problem over a finitely generated field of characteristic 0.
Using work of the first author [S. Bettin, High moments of the Estermann function. Algebra Number Theory47(3) (2018), 659–684], we prove a strong version of the Manin–Peyre conjectures with a full asymptotic and a power-saving error term for the two varieties respectively in
$\mathbb{P}^{2}\times \mathbb{P}^{2}$
with bihomogeneous coordinates
$[x_{1}:x_{2}:x_{3}],[y_{1}:y_{2},y_{3}]$
and in
$\mathbb{P}^{1}\times \mathbb{P}^{1}\times \mathbb{P}^{1}$
with multihomogeneous coordinates
$[x_{1}:y_{1}],[x_{2}:y_{2}],[x_{3}:y_{3}]$
defined by the same equation
$x_{1}y_{2}y_{3}+x_{2}y_{1}y_{3}+x_{3}y_{1}y_{2}=0$
. We thus improve on recent work of Blomer et al [The Manin–Peyre conjecture for a certain biprojective cubic threefold. Math. Ann.370 (2018), 491–553] and provide a different proof based on a descent on the universal torsor of the conjectures in the case of a del Pezzo surface of degree 6 with singularity type
$\mathbf{A}_{1}$
and three lines (the other existing proof relying on harmonic analysis by Chambert-Loir and Tschinkel [On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math.148 (2002), 421–452]). Together with Blomer et al [On a certain senary cubic form. Proc. Lond. Math. Soc.108 (2014), 911–964] or with work of the second author [K. Destagnol, La conjecture de Manin pour une famille de variétés en dimension supérieure. Math. Proc. Cambridge Philos. Soc.166(3) (2019), 433–486], this settles the study of the Manin–Peyre conjectures for this equation.