2,093 research outputs found
On Improving Roth's Theorem in the Primes
Let be a set of prime numbers containing
no non-trivial arithmetic progressions. Suppose that has relative density
, where denotes the number of primes in the set
. By modifying Helfgott and De Roton's work, we
improve their bound and show that Comment: 14 pages, to appear in Mathematik
Bounds For The Tail Distribution Of The Sum Of Digits Of Prime Numbers
Let s_q(n) denote the base q sum of digits function, which for n<x, is
centered around (q-1)/2 log_q x. In Drmota, Mauduit and Rivat's 2009 paper,
they look at sum of digits of prime numbers, and provide asymptotics for the
size of the set {p<x, p prime : s_q(p)=alpha(q-1)log_q x} where alpha lies in a
tight range around 1/2. In this paper, we examine the tails of this
distribution, and provide the lower bound |{p < x, p prime :
s_q(p)>alpha(q-1)log_q x}| >>x^{2(1-alpha)}e^{-c(log x)^{1/2+epsilon}} for
1/2<alpha<0.7375. To attain this lower bound, we note that the multinomial
distribution is sharply peaked, and apply results regarding primes in short
intervals. This proves that there are infinitely many primes with more than
twice as many ones than zeros in their binary expansion.Comment: 4 page
The Median Largest Prime Factor
Let denote the median largest prime factor of the integers in the
interval . We prove that
where . From this, we obtain the asymptotic
where is the Euler Mascheroni constant. This answers a
question posed by Martin, and improves a result of Selfridge and Wunderlich.Comment: 7 page
A Density Increment Approach to Roth's Theorem in the Primes
We prove that if is any set of prime numbers satisfying then must contain a -term arithmetic
progression. This is accomplished by combining the transference principle with
a density increment argument, exploiting the structure of the primes to obtain
a large density increase at each step of the iteration. The argument shows that
for any , and , if is a subset of primes contained in
with relative density at least then contains a -term
arithmetic progression.Comment: This has paper has been withdrawn due to an error in equation (2.8).
This error comes from the linearization step. I believe that the density
increment argument can be corrected, and a similar bound can be obtained by
moving entirely to Bohr sets. Currently this paper has a hole so I am
removing it from the arXi
Monochromatic Equilateral Triangles in the Unit Distance Graph
Let denote the minimum number of colors
needed to color so that there will not be a monochromatic
equilateral triangle with side length . Using the slice rank method, we
reprove a result of Frankl and Rodl, and show that
grows exponentially with . This
technique substantially improves upon the best known quantitative lower bounds
for , and we obtain Comment: 4 page
Upper bounds for sunflower-free sets
A collection of sets is said to form a -sunflower, or -system,
if the intersection of any two sets from the collection is the same, and we
call a family of sets sunflower-free if it contains no
sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt and
Croot, Lev and Pach we apply the polynomial method directly to
Erd\H{o}s-Szemer\'{e}di sunflower problem and prove that any sunflower-free
family of subsets of has size at most We say that
a set for is
sunflower-free if every distinct triple there exists a coordinate
where exactly two of are equal. Using a version of the
polynomial method with characters
instead of polynomials, we
show that any sunflower-free set has size
where . This can be
seen as making further progress on a possible approach to proving the
Erd\H{o}s-Rado sunflower conjecture, which by the work of Alon, Sphilka and
Umans is equivalent to proving that for some constant
independent of .Comment: 5 page
Primitive points in rational polygons
Let be a star-shaped polygon in the plane, with rational
vertices, containing the origin. The number of primitive lattice points in the
dilate is asymptotically Area as
. We show that the error term is both and . Both
bounds extend (to the above class of polygons) known results for the isosceles
right triangle, which appear in the literature as bounds for the error term in
the summatory function for Euler's .Comment: 17 page
Who will become dominant? Investigating the roles of individual behaviour, body size, and environmental predictability in brown trout fry hierarchies
This paper presents a study investigating performance of brown trout fry, with different behavioural characteristics, in environments differing in food predictability. Based on previous experimental findings, we hypothesised that more active individuals would be favoured by a predictable environment, as compared to an unpredictable environment, as a consequence of being more aggressive and likely to dominate the best feeding stations. This hypothesis was not supported, as more active individuals instead tended to perform better, in terms of growth and survival, in unpredictable environments. However, this effect may stem from initial size differences, as more active fish also tended to be larger. In predictable environments, no trends between activity (or size) and performance were detected. Dominant individuals could be identified based on lighter body colouration in 9 out of 10 rearing tanks, but dominance appeared not to be related to activity score. The results highlight a potential advantage of more active and/or larger fry in unpredictable environments, while performance in predictable environments is likely depending on other phenotypic characteristics. Our general experimental approach can be useful for further developments in the investigation of performance of different ethotypes of brown trout fry
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