Author

# Brian McDonald

Other affiliations: University of Chicago

Bio: Brian McDonald is an academic researcher from University of Rochester. The author has contributed to research in topics: Mathematics & Integer sequence. The author has an hindex of 3, co-authored 9 publications receiving 30 citations. Previous affiliations of Brian McDonald include University of Chicago.

##### Papers

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TL;DR: In this article, the distribution of the number of summands involved in such decompositions has been studied and it has been shown that the distribution converges to the standard normal.

Abstract: Zeckendorf's theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers ${F_n}$, with initial terms $F_1 = 1, F_2 = 2$. We consider the distribution of the number of summands involved in such decompositions. Previous work proved that as $n \to \infty$ the distribution of the number of summands in the Zeckendorf decompositions of $m \in [F_n, F_{n+1})$, appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in $[F_n, F_{n+1})$ share the same potential summands.
We generalize these results to subintervals of $[F_n, F_{n+1})$ as $n \to \infty$; the analysis is significantly more involved here as different integers have different sets of potential summands. Explicitly, fix an integer sequence $\alpha(n) \to \infty$. As $n \to \infty$, for almost all $m \in [F_n, F_{n+1})$ the distribution of the number of summands in the Zeckendorf decompositions of integers in the subintervals $[m, m + F_{\alpha(n)})$, appropriately normalized, converges to the standard normal. The proof follows by showing that, with probability tending to $1$, $m$ has at least one appropriately located large gap between indices in its decomposition. We then use a correspondence between this interval and $[0, F_{\alpha(n)})$ to obtain the result, since the summands are known to have Gaussian behavior in the latter interval. % We also prove the same result for more general linear recurrences.

17 citations

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TL;DR: In this article, it was shown that the distribution of the leading digits of the Fibonacci summands in its Zeckendorf decomposition converges to Benford's law almost surely.

Abstract: A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law if the density of the elements in $S$ with leading digit $d$ is $\log_{10}{(1+\frac{1}{d})}$; in other words, smaller leading digits are more likely to occur. We prove that, as $n\to\infty$, for a randomly selected integer $m$ in $[0, F_{n+1})$ the distribution of the leading digits of the Fibonacci summands in its Zeckendorf decomposition converge to Benford's law almost surely. Our results hold more generally, and instead of looking at the distribution of leading digits one obtains similar theorems concerning how often values in sets with density are attained.

8 citations

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TL;DR: In this article , it was shown that for sufficiently large subsets of Fq3, the Vapnik-Chervonenkis dimension of Ht3(E) is equal to 3.

4 citations

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Yale University

^{1}, University of Rochester^{2}, Williams College^{3}, Harvard University^{4}, North Dakota State University^{5}, University of Michigan^{6}TL;DR: In this paper, the authors combine mean value estimates of Montgomery and Vaughan with a method of Ramachandra to show that there are infinitely many gaps between consecutive zeros of L ( s, f ) along the critical line that are at most 3 = 1.732 times the average spacing.

3 citations

07 Oct 2022

TL;DR: It is proved, using the Weil bound for multiplicative character sums, that the Vapnik–Chervonenkis dimension of the set of quadratic residues, F q, is (cid:62) ( 12 + o (1) log 2 q .

Abstract: . We study the Vapnik–Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in ﬁnite ﬁelds, F q , when considered as a subset of the additive group. We conjecture that as q → ∞ , the squares have the maximum possible VC-dimension, viz. (1 + o (1)) log 2 q . We prove, using the Weil bound for multiplicative character sums, that the VC-dimension is (cid:62) ( 12 + o (1)) log 2 q . We also provide numerical evidence for our conjectures. The results generalize to multiplicative subgroups Γ ⊆ F × q of bounded index.

3 citations

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TL;DR: In this article, it was shown that for any positive linear recurrence, the probability of a gap larger than the recurrence length converges to decaying geometrically, and that the distribution of smaller gaps depends in a computable way on the coefficients of a recurrence.

19 citations

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TL;DR: In this article, the distribution of the number of summands involved in such decompositions has been studied and it has been shown that the distribution converges to the standard normal.

Abstract: Zeckendorf's theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers ${F_n}$, with initial terms $F_1 = 1, F_2 = 2$. We consider the distribution of the number of summands involved in such decompositions. Previous work proved that as $n \to \infty$ the distribution of the number of summands in the Zeckendorf decompositions of $m \in [F_n, F_{n+1})$, appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in $[F_n, F_{n+1})$ share the same potential summands.
We generalize these results to subintervals of $[F_n, F_{n+1})$ as $n \to \infty$; the analysis is significantly more involved here as different integers have different sets of potential summands. Explicitly, fix an integer sequence $\alpha(n) \to \infty$. As $n \to \infty$, for almost all $m \in [F_n, F_{n+1})$ the distribution of the number of summands in the Zeckendorf decompositions of integers in the subintervals $[m, m + F_{\alpha(n)})$, appropriately normalized, converges to the standard normal. The proof follows by showing that, with probability tending to $1$, $m$ has at least one appropriately located large gap between indices in its decomposition. We then use a correspondence between this interval and $[0, F_{\alpha(n)})$ to obtain the result, since the summands are known to have Gaussian behavior in the latter interval. % We also prove the same result for more general linear recurrences.

17 citations

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TL;DR: In this paper, it was shown that there are infinitely many consecutive zeros of the Riemann zeta function on the critical line whose gaps are greater than 3.18$ times the average spacing.

Abstract: We prove that there exist infinitely many consecutive zeros of the Riemann zeta-function on the critical line whose gaps are greater than $3.18$ times the average spacing. Using a modification of our method, we also show that there are even larger gaps between the multiple zeros of the zeta function on the critical line (if such zeros exist).

15 citations

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TL;DR: In this paper, a two-player game for linear recurrence relations of the form (G_n = \sum i=1}^{k} c G_{n-i}$ for a fixed positive integer $n$ and an initial decomposition of $n = n G_1$ was constructed, and the game always terminates in the Zeckendorf decomposition.

Abstract: Zeckendorf proved that every positive integer $n$ can be written uniquely as the sum of non-adjacent Fibonacci numbers; a similar result, though with a different notion of a legal decomposition, holds for many other sequences. We use these decompositions to construct a two-player game, which can be completely analyzed for linear recurrence relations of the form $G_n = \sum_{i=1}^{k} c G_{n-i}$ for a fixed positive integer $c$ ($c=k-1=1$ gives the Fibonaccis). Given a fixed integer $n$ and an initial decomposition of $n = n G_1$, the two players alternate by using moves related to the recurrence relation, and whomever moves last wins. The game always terminates in the Zeckendorf decomposition, though depending on the choice of moves the length of the game and the winner can vary. We find upper and lower bounds on the number of moves possible; for the Fibonacci game the upper bound is on the order of $n\log n$, and for other games we obtain a bound growing linearly with $n$. For the Fibonacci game, Player 2 has the winning strategy for all $n > 2$. If Player 2 makes a mistake on his first move, however, Player 1 has the winning strategy instead. Interestingly, the proof of both of these claims is non-constructive.

8 citations

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TL;DR: In this article, it was shown that the distribution of the leading digits of the Fibonacci summands in its Zeckendorf decomposition converges to Benford's law almost surely.

Abstract: A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law if the density of the elements in $S$ with leading digit $d$ is $\log_{10}{(1+\frac{1}{d})}$; in other words, smaller leading digits are more likely to occur. We prove that, as $n\to\infty$, for a randomly selected integer $m$ in $[0, F_{n+1})$ the distribution of the leading digits of the Fibonacci summands in its Zeckendorf decomposition converge to Benford's law almost surely. Our results hold more generally, and instead of looking at the distribution of leading digits one obtains similar theorems concerning how often values in sets with density are attained.

8 citations