scholarly journals SIMULTANEOUS ARITHMETIC PROGRESSIONS ON ALGEBRAIC CURVES

2011 ◽  
Vol 07 (04) ◽  
pp. 921-931 ◽  
Author(s):  
RYAN SCHWARTZ ◽  
JÓZSEF SOLYMOSI ◽  
FRANK DE ZEEUW
Keyword(s):  
Lower Bound ◽  
Elliptic Curve ◽  
Arithmetic Progression ◽  
Upper Bound ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Arithmetic Progressions ◽  
Real Algebraic Curve

A simultaneous arithmetic progression (s.a.p.) of length k consists of k points (xi, yσ(i)), where [Formula: see text] and [Formula: see text] are arithmetic progressions and σ is a permutation. Garcia-Selfa and Tornero asked whether there is a bound on the length of an s.a.p. on an elliptic curve in Weierstrass form over ℚ. We show that 4319 is such a bound for curves over ℝ. This is done by considering translates of the curve in a grid as a graph. A simple upper bound is found for the number of crossings and the "crossing inequality" gives a lower bound. Together these bound the length of an s.a.p. on the curve. We also extend this method to bound the k for which a real algebraic curve can contain k points from a k × k grid. Lastly, these results are extended to complex algebraic curves.

scholarly journals NONZERO VALUES OF DIRICHLET L-FUNCTIONS IN VERTICAL ARITHMETIC PROGRESSIONS

2013 ◽  
Vol 09 (04) ◽  
pp. 813-843 ◽  
Author(s):  
GREG MARTIN ◽  
NATHAN NG
Keyword(s):  
Arithmetic Progression ◽  
Positive Constant ◽  
Upper Bound ◽  
Arithmetic Progressions ◽  
Linearly Independent ◽  
Theoretical Evidence

Let L(s, χ) be a fixed Dirichlet L-function. Given a vertical arithmetic progression of T points on the line ℜs = ½, we show that at least cT/ log T of them are not zeros of L(s, χ) (for some positive constant c). This result provides some theoretical evidence towards the conjecture that all nonnegative ordinates of zeros of Dirichlet L-functions are linearly independent over the rationals. We also establish an upper bound (depending upon the progression) for the first member of the arithmetic progression that is not a zero of L(s, χ).

scholarly journals Coloring the Cube with Rainbow Cycles

10.37236/2957 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Dhruv Mubayi ◽  
Randall Stading
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Arithmetic Progression ◽  
Upper Bound ◽  
Sidon Sets ◽  
Dimensional Cube

For every even positive integer $k\ge 4$ let $f(n,k)$ denote the minimim number of colors required to color the edges of the $n$-dimensional cube $Q_n$, so that the edges of every copy of the $k$-cycle $C_k$ receive $k$ distinct colors. Faudree, Gyárfás, Lesniak and Schelp proved that $f(n,4)=n$ for $n=4$ or $n>5$. We consider larger $k$ and prove that if $k \equiv 0$ (mod 4), then there are positive constants $c_1, c_2$ depending only on $k$ such that$$c_1n^{k/4} < f(n,k) < c_2 n^{k/4}.$$Our upper bound uses an old construction of Bose and Chowla of generalized Sidon sets. For $k \equiv 2$ (mod 4), the situation seems more complicated. For the smallest case $k=6$ we show that $$3n-2 \le f(n, 6) < n^{1+o(1)}$$ with the lower bound holding for $n \ge 3$. The upper bound is obtained from Behrend's construction of a subset of integers with no three term arithmetic progression.

Rational Points in Arithmetic Progressions on y2 = xn + k

2012 ◽  
Vol 55 (1) ◽  
pp. 193-207 ◽  
Author(s):  
Maciej Ulas
Keyword(s):  
Elliptic Curve ◽  
Arithmetic Progression ◽  
Hyperelliptic Curve ◽  
Rational Points ◽  
Arithmetic Progressions ◽  
Multiple Roots

AbstractLet C be a hyperelliptic curve given by the equation y2 = f(x) for f ∈ ℤ[x] without multiple roots. We say that points Pi = (xi, yi) ∈ C(ℚ) for i = 1, 2, … , m are in arithmetic progression if the numbers xi for i = 1, 2, … , m are in arithmetic progression.In this paper we show that there exists a polynomial k ∈ ℤ[t] with the property that on the elliptic curve ε′ : y2 = x3+k(t) (defined over the field ℚ(t)) we can find four points in arithmetic progression that are independent in the group of all ℚ(t)-rational points on the curve Ε′. In particular this result generalizes earlier results of Lee and Vélez. We also show that if n ∈ ℕ is odd, then there are infinitely many k's with the property that on curves y2 = xn + k there are four rational points in arithmetic progressions. In the case when n is even we can find infinitely many k's such that on curves y2 = xn +k there are six rational points in arithmetic progression.

scholarly journals A new proof of Halász’s theorem, and its consequences

2018 ◽  
Vol 155 (1) ◽  
pp. 126-163 ◽  
Author(s):  
Andrew Granville ◽  
Adam J. Harper ◽  
K. Soundararajan
Keyword(s):  
Current Literature ◽  
Arithmetic Progression ◽  
Upper Bound ◽  
Multiplicative Function ◽  
Mean Value ◽  
Arithmetic Progressions ◽  
Short Intervals ◽  
The Mean

Halász’s theorem gives an upper bound for the mean value of a multiplicative function$f$. The bound is sharp for general such$f$, and, in particular, it implies that a multiplicative function with$|f(n)|\leqslant 1$has either mean value$0$, or is ‘close to’$n^{it}$for some fixed$t$. The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to treat short intervals and arithmetic progressions. We use these results to obtain new, arguably simpler, proofs that there are always primes in short intervals (Hoheisel’s theorem), and that there are always primes near to the start of an arithmetic progression (Linnik’s theorem).

scholarly journals Improved bounds for arithmetic progressions in product sets

2015 ◽  
Vol 11 (08) ◽  
pp. 2295-2303 ◽  
Author(s):  
Dmitrii Zhelezov
Keyword(s):  
Lower Bound ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Complex Numbers ◽  
Natural Numbers ◽  
Multiplicative Constant ◽  
Product Sets

Let B be a set of natural numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = {bb′|b, b′ ∈ B} cannot be greater than O(n log n) which matches the lower bound provided in an earlier paper up to a multiplicative constant. For sets of complex numbers, we improve the bound to Oϵ(n1 + ϵ) for arbitrary ϵ > 0 assuming the GRH.

scholarly journals The Barban-Vehov Theorem in Arithmetic Progressions

2019 ◽  
Author(s):  
V Kumar Murty
Keyword(s):  
Asymptotic Formula ◽  
Arithmetic Progression ◽  
Upper Bound ◽  
Arithmetic Progressions ◽  
Mean Square ◽  
Titchmarsh Theorem ◽  
International Audience ◽  
The Mean ◽  
Selberg's Sieve

International audience A result of Barban-Vehov (and independently Motohashi) gives an estimate for the mean square of a sequence related to Selberg's sieve. This upper bound was refined to an asymptotic formula by S. Graham in 1978. In 1992, I made the observation that Graham's method can be used to obtain an asymptotic formula when the sum is restricted to an arithmetic progression. This formula immediately gives a version of the Brun-Titchmarsh theorem. I am taking the occasion of a volume in honour of my friend S. Srinivasan to revisit and publish this observation in the hope that it might still be of interest.

scholarly journals Distinct Distances on Algebraic Curves in the Plane

2016 ◽  
Vol 26 (1) ◽  
pp. 99-117 ◽  
Author(s):  
JÁNOS PACH ◽  
FRANK DE ZEEUW
Keyword(s):  
Lower Bound ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Parallel Lines ◽  
Plane Algebraic Curve ◽  
Concentric Circles

LetSbe a set ofnpoints in${\mathbb R}^{2}$contained in an algebraic curveCof degreed. We prove that the number of distinct distances determined bySis at leastcdn4/3, unlessCcontains a line or a circle.We also prove the lower boundcd′ min{m2/3n2/3,m2,n2} for the number of distinct distances betweenmpoints on one irreducible plane algebraic curve andnpoints on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer and Solymosi in [19].

scholarly journals Minimum Number of Colours to Avoid k-Term Monochromatic Arithmetic Progressions

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 247
Author(s):  
Kai An Sim ◽  
Kok Bin Wong
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Upper Bound ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Arithmetic Progressions ◽  
Minimum Number ◽  
Necessary And Sufficient

By recalling van der Waerden theorem, there exists a least a positive integer w=w(k;r) such that for any n≥w, every r-colouring of [1,n] admits a monochromatic k-term arithmetic progression. Let k≥2 and rk(n) denote the minimum number of colour required so that there exists a rk(n)-colouring of [1,n] that avoids any monochromatic k-term arithmetic progression. In this paper, we give necessary and sufficient conditions for rk(n+1)=rk(n). We also show that rk(n)=2 for all k≤n≤2(k−1)2 and give an upper bound for rp(pm) for any prime p≥3 and integer m≥2.

scholarly journals The number of connected components of certain real algebraic curves

2001 ◽  
Vol 25 (11) ◽  
pp. 693-701
Author(s):  
Seon-Hong Kim
Keyword(s):  
Algebraic Curve ◽  
Algebraic Curves ◽  
Connected Components ◽  
Connected Component ◽  
The Real ◽  
Real Algebraic Curves ◽  
Real Algebraic Curve

For an integern≥2, letp(z)=∏k=1n(z−αk)andq(z)=∏k=1n(z−βk), whereαk,βkare real. We find the number of connected components of the real algebraic curve{(x,y)∈ℝ2:|p(x+iy)|−|q(x+iy)|=0}for someαkandβk. Moreover, in these cases, we show that each connected component contains zeros ofp(z)+q(z), and we investigate the locus of zeros ofp(z)+q(z).

scholarly journals Discrepancy of Cartesian Products of Arithmetic Progressions

10.37236/1758 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Benjamin Doerr ◽  
Anand Srivastav ◽  
Petra Wehr
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Arithmetic Progressions ◽  
Cartesian Products ◽  
Fixed Integer ◽  
Higher Dimensional ◽  
Special Cases ◽  
Dimensional Version ◽  
Compact Abelian Groups ◽  
Combinatorial Discrepancy

We determine the combinatorial discrepancy of the hypergraph ${\cal H}$ of cartesian products of $d$ arithmetic progressions in the $[N]^d$–lattice ($[N] = \{0,1,\ldots,N-1\}$). The study of such higher dimensional arithmetic progressions is motivated by a multi-dimensional version of van der Waerden's theorem, namely the Gallai-theorem (1933). We solve the discrepancy problem for $d$–dimensional arithmetic progressions by proving ${\rm disc}({\cal H}) = \Theta(N^{d/4})$ for every fixed integer $d \ge 1$. This extends the famous lower bound of $\Omega(N^{1/4})$ of Roth (1964) and the matching upper bound $O(N^{1/4})$ of Matoušek and Spencer (1996) from $d=1$ to arbitrary, fixed $d$. To establish the lower bound we use harmonic analysis on locally compact abelian groups. For the upper bound a product coloring arising from the theorem of Matoušek and Spencer is sufficient. We also regard some special cases, e.g., symmetric arithmetic progressions and infinite arithmetic progressions.

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