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Abstract We study the extent to which divisors of a typical integer n are concentrated. In particular, defining $$\Delta (n) := \max _t \# \{d | n, \log d \in [t,t+1]\}$$ Δ ( n ) : = max t # { d | n , log d ∈ [ t , t + 1 ] } , we show that $$\Delta (n) \geqslant (\log \log n)^{0.35332277\ldots }$$ Δ ( n ) ⩾ ( log log n ) 0.35332277 … for almost all n , a bound we believe to be sharp. This disproves a conjecture of Maier and Tenenbaum. We also prove analogs for the concentration of divisors of a random permutation and of a random polynomial over a finite field. Most of the paper is devoted to a study of the following much more combinatorial problem of independent interest. Pick a random set $${\textbf{A}} \subset {\mathbb {N}}$$ A ⊂ N by selecting i to lie in $${\textbf{A}}$$ A with probability 1/ i . What is the supremum of all exponents $$\beta _k$$ β k such that, almost surely as $$D \rightarrow \infty $$ D → ∞ , some integer is the sum of elements of $${\textbf{A}} \cap [D^{\beta _k}, D]$$ A ∩ [ D β k , D ] in k different ways? We characterise $$\beta _k$$ β k as the solution to a certain optimisation problem over measures on the discrete cube $$\{0,1\}^k$$ { 0 , 1 } k , and obtain lower bounds for $$\beta _k$$ β k which we believe to be asymptotically sharp.
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A Strong-Type Furstenberg–Sárközy Theorem for Sets of Positive Measure
Abstract For every $$\beta \in (0,\infty )$$ β ∈ ( 0 , ∞ ) , $$\beta \ne 1$$ β ≠ 1 , we prove that a positive measure subset A of the unit square contains a point $$(x_0,y_0)$$ ( x 0 , y 0 ) such that A nontrivially intersects curves $$y-y_0 = a (x-x_0)^\beta $$ y - y 0 = a ( x - x 0 ) β for a whole interval $$I\subseteq (0,\infty )$$ I ⊆ ( 0 , ∞ ) of parameters $$a\in I$$ a ∈ I . A classical Nikodym set counterexample prevents one to take $$\beta =1$$ β = 1 , which is the case of straight lines. Moreover, for a planar set A of positive density, we show that the interval I can be arbitrarily large on the logarithmic scale. These results can be thought of as Bourgain-style large-set variants of a recent continuous-parameter Sárközy-type theorem by Kuca, Orponen, and Sahlsten.
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- Award ID(s):
- 2154356
- PAR ID:
- 10425149
- Date Published:
- Journal Name:
- The Journal of Geometric Analysis
- Volume:
- 33
- Issue:
- 8
- ISSN:
- 1050-6926
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s12220-023-01309-7
