We don't really know the extent of territory that might have been occupied by the population represented by the Denisova genome. The signs of mixture into the Melanesian/New Guinea population suggests that the Denisova individual shared many genes with people who lived somewhere along the South or Southeast Asian coast. Denisova itself, however, is in the Altai Mountains.
Last week I wrote some thoughts about the possible introgression of HLA alleles from Denisovans into more recent populations. HLA genes pose many problems for testing this hypothesis -- including the difficulty of identifying the alleles in a low-coverage genome and the high chance of incomplete lineage sorting of ancient alleles in recent populations. Other parts of the genome in principle may be much easier to find evidence of introgression.
If an allele that originated in Denisovans had some advantage in later populations, it might today be found very widely spread across Asian populations, even if the amount of Denisovan ancestry in most of these populations is very small. This was the theme of my paper with Gregory Cochran several years ago  ("The inevitability of introgression"). The probability that a single copy of an advantageous allele will survive and increase in the population is roughly 2s, where s is the fitness advantage in a heterozygote carrying the allele. A relatively small number of copies of an allele might have entered a recent human population by introgression from some ancient population, but these few copies would have a high likelihood of surviving and increasing in frequency, possibly toward fixation. HLA alleles could easily be in this category, but the challenges identifying them and high chance of ILS make the hypothesis hard to test.
Another strategy is to identify genes that have been selected in recent populations and see if the linked haplotype shows up in the Denisova genome. Recently, several studies have attempted to identify genes related to high altitude adaptation in Tibetans. At least some Denisovans lived in the mountainous areas of central Asia, and so I'm curious whether they might have some alleles adapted to this environment. The Altai are not nearly as high as the Tibetan plateau (in fact Denisova itself is not much higher than western Kansas), and we don't know how long Denisovan people might have been resident in Central Asia, but if we're looking for selected alleles there are some strong candidates in this category of genes.
So let's look at some of them. All positions here are mapped to the hg18 human genome assembly.
Yi and colleagues  find a strong frequency difference between China and Tibet for a SNP in EPAS1, at chr2:46441523. The derived allele, G, has a frequency of 87% in their Tibetan sample but only 9% in their Chinese sample (and zero in Denmark). The Denisova genome is represented by two reads at this site, both C, the ancestral allele. We don't necessarily have to accept that this is a functional site, but as the marker most strongly differentiating the high altitude population it would likely be closely linked to any functional variant. So the Denisova allele suggests that this ancient individual lacked whatever functional variant might currently be common in Tibetans for this gene.
Simonson and colleagues  took a different approach, focusing on candidate genes that they argued a priori were likely to be involved in adaptation to hypoxia because of their physiological role. They evaluated these genes for evidence of positive selection in Tibetans, finding several candidate haplotypes for recent adaptive evolution to high altitude.
For each of five genes, they identified a three-locus "core selection haplotype" that shows signs of selection within Tibet. The purpose of these three-SNP haplotypes was to examine the correlation of haplotypes and phenotypes in a sample of people where physiological data were taken. So they are intended as tags, not as comprehensive and unique identifiers of the candidates at the genetic level. But the three-locus haplotypes are the only ones reported in the supplement to the paper, so that's what I have to compare.
EGLN1: The three-allele candidate selected haplotype consists of A at chr1:229793717, T at chr1:229667980 and T at chr1:229665156. Denisova apparently has the selected haplotype with A at chr1:229793717 (2/2 reads), T at chr1:229667980 (3/3 reads) and T at chr1:229665156 (1/1 reads). However, it is not obvious whether this is significant. All three alleles on the candidate selected haplotype are the ancestral (present in chimpanzees and gorillas) alleles, which are much more likely to show up in the archaic genomes than derived alleles. These ancestral alleles are also present in several of the whole genomes provided along with the Denisova sequence reads. So it's not clear to me how good a candidate for selection the haplotype really is.
CYP17A1: Here the three-allele candidate selected haplotype includes G at chr10:104568521, G at chr10:104594906, and C at chr10:104517420. Denisova has C (5/5 reads, ancestral), T (4/4 reads, ancestral), and C (3/3 reads, ancestral). Again, Denisova has the all-ancestral haplotype here, but in this case it is not the selection candidate.
PTEN: The selected candidate haplotype is G at chr10:89770364, C at chr10:89790851 and C at chr10:89778618. Denisova has G (5/5 reads, ancestral), T (2/2 reads, derived), and C (4/4 reads, ancestral). Not selected.
I always find it interesting when the Denisova genome has a derived allele at an interesting site -- it is the shared derived alleles between these archaic genomes and living people that constitute evidence of genetic persistence of the archaic people. No single site carries that information (any one allele may be shared by incomplete lineage sorting), but I still like to note them. The Papuan and half the Native American, Sardinian and Mongolian reads share the derived T at chr10:89790851 with Denisova.
HMOX2: The candidate selected haplotype has C at chr16:4456093, T at chr16:4465266, T at chr16:4442515. Denisova has this candidate selected haplotype: C (3/3 reads, ancestral), T (4/4 reads, ancestral), T (5/5 reads, ancestral). That haplotype may also be in the Cambodian whole genome accompanying the Denisova data, and can't be ruled out for the Mongolian. Again, the all-ancestral haplotype and wider distribution argue against the hypothesis that this haplotype was specifically selected in Tibet.
PPARA: The core candidate selected haplotype has A at chr22:44827140, C at chr22:44832376 and T at chr22:44842095. Denisova has A (8/8 reads, ancestral), A (5/5 reads, ancestral), and C (2/2 reads, ancestral). Notice again, Denisova has the all-ancestral haplotype. As an ancient sequence, we are finding this is the usual case, human-derived alleles are just rarer in this genome.
OK, where are we? Out of six genes that are candidates for selection on altitude adaptation in Tibetans, the Denisova genome has two -- at ELGN1 and HMOX2. In both cases, the core selected haplotype consists entirely of ancestral alleles, and so I think they are actually poor evidence of introgression on the surface. I would test them by looking at more SNPs linked to the presumed selected haplotype, hoping to find some derived SNPs shared by the Denisovan genome and the presumed selected haplotypes. Unfortunately, publications do not yet routinely report long haplotypes, so it will take some more digging to test these cases.