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Kevin O'Bryant

Publications and source records attributed to Kevin O'Bryant.

At least 37 records · Page 2Linked to original sources
arXiv2013

Irrational numbers associated to sequences without geometric progressions

Let s and k be integers with s \geq 2 and k \geq 2. Let g_k^{(s)}(n) denote the cardinality of the largest subset of the set {1,2,..., n} that contains no geometric progression of length k whose common ratio is a power of s. Let r_k(\ell) denote the cardinality of the largest subset of the set {0,1,2,\ldots, \ell -1\} that contains no arithmetric progression of length k. The limit \[ \lim_{n\rightarrow \infty} \frac{g_k^{(s)}(n)}{n} = (s-1) \sum_{m=1}^{\infty} \left(\frac{1}{s} \right)^{\min \left(r_k^{-1}(m)\right)} \] exists and converges to an irrational number.

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arXiv2012

Distribution of Missing Sums in Sumsets

For any finite set of integers X, define its sumset X+X to be {x+y: x, y in X}. In a recent paper, Martin and O'Bryant investigated the distribution of |A+A| given the uniform distribution on subsets A of {0, 1, ..., n-1}. They also conjectured the existence of a limiting distribution for |A+A| and showed that the expectation of |A+A| is 2n - 11 + O((3/4)^{n/2}). Zhao proved that the limits m(k) := lim_{n --> oo} Prob(2n-1-|A+A|=k) exist, and that sum_{k >= 0} m(k)=1. We continue this program and give exponentially decaying upper and lower bounds on m(k), and sharp bounds on m(k) for small k. Surprisingly, the distribution is at least bimodal; sumsets have an unexpected bias against missing exactly 7 sums. The proof of the latter is by reduction to questions on the distribution of related random variables, with large scale numerical computations a key ingredient in the analysis. We also derive an explicit formula for the variance of |A+A| in terms of Fibonacci numbers, finding Var(|A+A|) is approximately 35.9658. New difficulties arise in the form of weak dependence between events of the form {x in A+A}, {y in A+A}. We surmount these obstructions by translating the problem to graph theory. This approach also yields good bounds on the probability for A+A missing a consecutive block of length k.

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arXiv2010

Thick subsets that do not contain arithmetic progressions

We adapt the construction of subsets of {1, 2, ..., N} that contain no k-term arithmetic progressions to give a relatively thick subset of an arbitrary set of N integers. Particular examples include a thick subset of {1, 4, 9, ..., N^2} that does not contain a 3-term AP, and a positive relative density subset of a random set (contained in {1, 2, ..., n} and having density c n^{-1/(k-1)}) that is free of k-term APs.

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arXiv2010

Sets of integers that do not contain long arithmetic progressions

In 1946, Behrend gave a construction of dense finite sets of integers that do not contain 3-term arithmetic progressions. In 1961, Rankin generalized Behrend's construction to sets avoiding k-term arithmetic progressions, and in 2008 Elkin refined Behrend's 3-term construction. In this work, we combine Elkin's refinement and Rankin's generalization. Arithmetic progressions are handled as a special case of polynomial progressions. In 1946, Behrend gave a construction of dense finite sets of integers that do not contain a 3-term arithmetic progression (AP). In 1961, Rankin generalized Behrend's construction to sets avoiding k-term APs. In 2008, Elkin refined Behrend's 3-term construction, and later in 2008, Green & Wolf found a distinct approach (albeit morally similar) that is technically more straightforward. This work combines Elkin's refinement and Rankin's generalization in the Green & Wolf framework. A curious aspect of the construction is that we induct through sets that do not contain a long polynomial progression in order to construct a set without a long AP. The bounds for r_k(N), the largest size of a subset of {1,2,...,N} that does not contain a k element AP, are (where \log=\log_2, for sufficiently large N, with n=\ceiling{\log k}): r_3(N) > N (\sqrt{360}/(e π^{3/2})-ε) \sqrt[4]{2\log N} * 4^{-\sqrt{2 \log N}}, r_k(N) > CN 2^{-n 2^{(n-1)/2} \sqrt[n]{\log N}+\frac{1}{2n}\log\log N}. The improvement over earlier work is in the simplification of the construction, the explicitness of the bound for r_3, and in the \log\log term for general k.

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arXiv2009

The supremum of autoconvolutions, with applications to additive number theory

We adapt a number-theoretic technique of Yu to prove a purely analytic theorem: if f(x) is in L^1 and L^2, is nonnegative, and is supported on an interval of length I, then the supremum of the convolution f*f is at least 0.631 \| f \|_1^2 / I. This improves the previous bound of 0.591389 \| f \|_1^2 / I. Consequently, we improve the known bounds on several related number-theoretic problems. For a subset A of {1,2, ..., n}, let g be the maximum multiplicity of any element of the multiset {a+b: a,b in A}. Our main corollary is the inequality gn>0.631|A|^2, which holds uniformly for all g, n, and A.

math.CA↗
arXiv2008

Can you hear the shape of a Beatty sequence?

Let K(x_1,...,x_d) be a polynomial. If you are not given the real numbers α_1, α_2, ...,α_d, but are given the polynomial K and the sequence a_n=K(\floor{nα_1},\floor{nα_2},...,\floor{nα_d}), can you deduce the values of α_i? Not, it turns out, in general. But with additional irrationality hypotheses and certain polynomials, it is possible. We also consider the problem of deducing α_i from the integer sequence with nested flooring (\floor{\floor{... \floor{\floor{nα_1}α_2}... α_{d-1}}α_d})_{n=1}^\infty.

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arXiv2007

Gaps in the Spectrum of Heights of Projective Points

Let r mod m be the least positive residue of r modulo m, and set the height of a pair (r,s) of integers, both relatively prime to m, to be the minimum over k, with 0 infinity, gcd(r_i,m_i)=gcd(s_i,m_i)=1, and m_i^{-1} h(m_i,r_i,s_i) --> x. The main result here is that SPECTRUM is the union of {0} and {1/k : k =1,2,...}.

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arXiv2007

Binary linear forms over finite sets of integers

Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u_1x+v_1y and g(x,y)=u_2x+v_2y with integral coefficients, there exist arbitrarily large finite sets of integers A and B such that |f(A)| > |g(A)| and |f(B)| < |g(B)|.

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arXiv2006

Many sets have more sums than differences

Since addition is commutative but subtraction is not, the sumset S+S of a finite set S is predisposed to be smaller than the difference set S-S. In this paper, however, we show that each of the three possibilities (|S+S|>|S-S|, |S+S|=|S-S|, |S+S|<|S-S|) occur for a positive proportion of the subsets of {0, 1, ..., n-1}. We also show that the difference |S+S| - |S-S| can take any integer value, and we show that the expected number of omitted differences is asymptotically 6 while the expected number of missing sums is asymptotically 10. Other data and conjectures on the distribution of these quantities are also given.

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arXiv2006

On the Sum of the Heights of Sturmian Factors

A binary word is a map W : N --> {0,1}, and the set of factors of W with length n is F_n(W):={(W(i),W(i+1),...,W(i+n-1)) : i >= 0}. A word is Sturmian if |F_n(W)|=n+1 for every n>0. We show that the sum of the heights (also known as hamming weights) of the n+1 factors with length n of a binary Sturmian word has the same parity as n, independent of W.

math.CO↗
arXiv2006

The Symmetric Subset Problem in Continuous Ramsey Theory

A symmetric subset of the reals is one that remains invariant under some reflection z --> c-z. We consider, for any 0 < x <= 1, the largest real number D(x) such that every subset of $[0,1]$ with measure greater than x contains a symmetric subset with measure D(x). In this paper we establish upper and lower bounds for D(x) of the same order of magnitude: for example, we prove that D(x) = 2x - 1 for 11/16 <= x <= 1 and that 0.59 x^2 < D(x) < 0.8 x^2 for 0 < x <= 11/16. This continuous problem is intimately connected with a corresponding discrete problem. A set S of integers is called a B*[g] set if for any given m there are at most g ordered pairs (s_1,s_2) \in S \times S with s_1+s_2 = m; in the case g=2, these are better known as Sidon sets. Our lower bound on D(x) implies that every B*[g] set contained in \{1,2,...,n\} has cardinality less than 1.30036 \sqrt{gn}. This improves a result of Green for g >= 30. Conversely, we use a probabilistic construction of B*[g] sets to establish an upper bound on D(x) for small x.

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arXiv2005

Almost Alternating Sums

Writing for a general mathematical audience, we provide elementary upper and lower bounds on the growth (as a function of N) of the sum \sum_{n=1}^N (-1)^{\floor{n x}} for various fixed x. For example, if x is a quadratic irrational, then the sum is O(log N), and if x is 2/(e-1), then the sum is O(log N / log log N). We compute the optimal big-Oh constant for x=\sqrt{2}, 1+\sqrt{5}, 2+\sqrt{10}, ....

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arXiv2005

Reciprocals of Binary Power Series

If A is a set of nonnegative integers containing 0, then there is a unique nonempty set B of nonnegative integers such that every positive integer can be written in the form a+b, where a\in A and b\in B, in an even number of ways. We compute the natural density of B for several specific sets A, including the Prouhet-Thue-Morse sequence, {0} \cup {2^n : n \geq 0}, and random sets, and we also study the distribution of densities of B for finite sets A. This problem is motivated by Euler's observation that if A is the set of n that have an odd number of partitions, then B is the set of pentagonal numbers {n(3n+1)/2 : n \in Z}. We also elaborate the connection between this problem and the theory of de Bruijn sequences and linear shift registers.

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arXiv2005

A Discrete Fourier Kernel and Fraenkel's Tiling Conjecture

The set B_{p,r}^q:=\{\floor{nq/p+r} \colon n\in Z \} with integers p, q, r) is a Beatty set with density p/q. We derive a formula for the Fourier transform \hat{B_{p,r}^q}(j):=\sum_{n=1}^p e^{-2 πi j \floor{nq/p+r} / q}. A. S. Fraenkel conjectured that there is essentially one way to partition the integers into m>2 Beatty sets with distinct densities. We conjecture a generalization of this, and use Fourier methods to prove several special cases of our generalized conjecture.

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arXiv2005

Constructions of Generalized Sidon Sets

We give explicit constructions of sets S with the property that for each integer k, there are at most g solutions to k=s_1+s_2, s_i\in S; such sets are called Sidon sets if g=2 and generalized Sidon sets if g\ge 3. We extend to generalized Sidon sets the Sidon-set constructions of Singer, Bose, and Ruzsa. We also further optimize Koulantzakis' idea of interleaving several copies of a Sidon set, extending the improvements of Cilleruelo & Ruzsa & Trujillo, Jia, and Habsieger & Plagne. The resulting constructions yield the largest known generalized Sidon sets in virtually all cases.

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arXiv2004

A Complete Annotated Bibliography of Work Related to Sidon Sequences

A Sidon sequence is a sequence of integers a_1 < a_2 < a_3 < ... with the property that the sums a_i+a_j (i\le j) are distinct. This work contains a survey of Sidon sequences and their generalizations, and an extensive annotated and hyperlinked bibliography of related work.

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