Prescribing the binary digits of squarefree numbers and quadratic residues

We study the equidistribution of multiplicatively defined sets, such as the squarefree integers, quadratic non-residues or primitive roots, in sets which are described in an additive way, such as sumsets or Hilbert cubes. In particular, we show that if one fixes any proportion less than $40\%$ of the digits of all numbers of a given binary bit length, then the remaining set still has the asymptotically expected number of squarefree integers. Next, we investigate the distribution of primitive roots modulo a large prime $p$, establishing a new upper bound on the largest dimension of a Hilbert cube in the set of primitive roots, improving on a previous result of the authors. Finally, we study sumsets in finite fields and asymptotically find the expected number of quadratic residues and non-residues in such sumsets, given their cardinalities are big enough. This significantly improves on a recent result by Dartyge, Mauduit and S\'ark\"ozy. Our approach introduces several new ideas, combining a variety of methods, such as bounds of exponential and character sums, geometry of numbers and additive combinatorics

Hadamard partitioned difference families and their descendants

If $D$ is a $(4u^2,2u^2-u,u^2-u)$ Hadamard difference set (HDS) in $G$, then $\{G,G\setminus D\}$ is clearly a $(4u^2,[2u^2-u,2u^2+u],2u^2)$ partitioned difference family (PDF). Any $(v,K,\lambda)$-PDF will be said of Hadamard-type if $v=2\lambda$ as the one above. We present a doubling construction which, starting from any such PDF, leads to an infinite class of PDFs. As a special consequence, we get a PDF in a group of order $4u^2(2n+1)$ and three block-sizes $4u^2-2u$, $4u^2$ and $4u^2+2u$, whenever we have a $(4u^2,2u^2-u,u^2-u)$-HDS and the maximal prime power divisors of $2n+1$ are all greater than $4u^2+2u$

New 22-designs from strong difference families

Strong difference families are an interesting class of discrete structures which can be used to derive relative difference families. Relative difference families are closely related to $2$-designs, and have applications in constructions for many significant codes, such as optical orthogonal codes and optical orthogonal signature pattern codes. In this paper, with a careful use of cyclotomic conditions attached to strong difference families, we improve the lower bound on the asymptotic existence results of $(\mathbb{F}_{p}\times \mathbb{F}_{q},\mathbb{F}_{p}\times \{0\},k,\lambda)$-DFs for $k\in\{p,p+1\}$. We improve Buratti's existence results for $2$-$(13q,13,\lambda)$ designs and $2$-$(17q,17,\lambda)$ designs, and establish the existence of seven new $2$-$(v,k,\lambda)$ designs for $(v,k,\lambda)\in\{(694,7,2),(1576,8,1),(2025,9,1),(765,9,2),(1845,9,2),(459,9,4)$, $(783,9,4)\}$.Comment: Version 1 is named "Improved cyclotomic conditions leading to new 2-designs: the use of strong difference families". Major revision according to the referees' comment

Frame difference families and resolvable balanced incomplete block designs

Frame difference families, which can be obtained via a careful use of cyclotomic conditions attached to strong difference families, play an important role in direct constructions for resolvable balanced incomplete block designs. We establish asymptotic existences for several classes of frame difference families. As corollaries new infinite families of 1-rotational $(pq+1,p+1,1)$-RBIBDs over $\mathbb{F}_{p}^+ \times \mathbb{F}_{q}^+$ are derived, and the existence of $(125q+1,6,1)$-RBIBDs is discussed. We construct $(v,8,1)$-RBIBDs for $v\in\{624,1576,2976,5720,5776,10200,14176,24480\}$, whose existence were previously in doubt. As applications, we establish asymptotic existences for an infinite family of optimal constant composition codes and an infinite family of strictly optimal frequency hopping sequences.Comment: arXiv admin note: text overlap with arXiv:1702.0750